oeis_superseeker_answer.txt

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SUGGESTION: IT APPEARS THAT THE SEQUENCE OF ABSOLUTE VALUES
CAN BE OBTAINED FROM THE FOLLOWING SEQUENCE(S) IN THE ENCYCLOPEDIA
(UP TO A LIMIT OF 50) BY 2 INSERTIONS, DELETIONS OR SUBSTITUTIONS:
 

%I A000010 M0299 N0111
%S A000010 1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,12,18,12,
%T A000010 28,8,30,16,20,16,24,12,36,18,24,16,40,12,42,20,24,22,46,16,42,20,32,
%U A000010 24,52,18,40,24,36,28,58,16,60,30,36,32,48,20,66,32,44
%N A000010 Euler totient function phi(n): count numbers <= n and prime to n.
%C A000010 Number of elements in a reduced residue system modulo n.
%C A000010 Degree of the n-th cyclotomic polynomial (cf. A013595). - _Benoit Cloitre_, Oct 12 2002
%C A000010 Number of distinct generators of a cyclic group of order n. Number of primitive n-th roots of unity. (A primitive n-th root x is such that x^k is not equal to 1 for k = 1, 2, ..., n - 1, but x^n = 1.) - _Lekraj Beedassy_, Mar 31 2005
%C A000010 Also number of complex Dirichlet characters modulo n; sum(k = 1, n, a(k)) is asymptotic to (3/Pi^2)*n^2. - _Steven Finch_, Feb 16 2006
%C A000010 a(n) is the highest degree of irreducible polynomial dividing 1 + x + x^2 + ... + x^(n-1) = (x^n - 1)/(x - 1). - _Alexander Adamchuk_, Sep 02 2006, corrected Sep 27 2006
%C A000010 a(p) = p - 1 for prime p. a(n) is even for n > 2. For n > 2 a(n)/2 = A023022(n) = number of partitions of n into 2 ordered relatively prime parts. - _Alexander Adamchuk_, Jan 25 2007
%C A000010 Number of automorphisms of the cyclic group of order n. - _Benoit Jubin_, Aug 09 2008
%C A000010 a(n+2) equals the number of palindromic Sturmian words of length n which are 'bispecial', prefix or suffix of two Sturmian words of length n + 1. - _Fred Lunnon_, Sep 05 2010
%C A000010 Suppose that a and n are coprime positive integers, then by Euler's totient theorem, any factor of n divides a^phi(n) - 1. - _Lei Zhou_, Feb 28 2012
%C A000010 a(n) = A096396(n) + A096397(n). - _Reinhard Zumkeller_, Mar 24 2012
%C A000010 If m has k prime factors,(f1, f2, ..., fk), then phi(m*n) = phi(f1*n) * phi(f2*n) * ... * phi(fk*n)/phi(n)^(k-1). For example, phi(42*n) = phi(2*n) * phi(3*n) * phi(7*n)/phi(n)^2. - _Gary Detlefs_, Apr 21 2012
%C A000010 sum(n>=1, a(n)/n! ) = 1.954085357876006213144... This sum is referenced in Plouffe's inverter. - _Alexander R. Povolotsky_, Feb 02 2013
%C A000010 The order of the multiplicative group of units modulo n. - _Michael Somos_, Aug 27 2013
%D A000010 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A000010 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
%D A000010 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.
%D A000010 C. W. Curtis, Pioneers of Representation Theory ..., Amer. Math. Soc., 1999; see p. 3.
%D A000010 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, Chapter V.
%D A000010 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
%D A000010 Carl Friedrich Gauss, "Disquitiones Arithmeticae", Yale University Press, 1965; see p. 21.
%D A000010 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, p. 137.
%D A000010 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 60, 62, 63, 288, 323, 328, 330.
%D A000010 Peter Hilton and Jean Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, (pages 261-264, the Coach theorem)
%D A000010 P. Ribenboim, The New Book of Prime Number Records.
%D A000010 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000010 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000010 N. J. A. Sloane and Daniel Forgues, <a href="/A000010/b000010.txt">Table of n, phi(n) for n=1..100000</a> (first 10000 terms from N. J. A. Sloane)
%H A000010 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.7, pp. 776-778
%H A000010 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A000010 D. Alpern, <a href="http://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method(along with sigma_0, sigma_1 and phi functions)</a>
%H A000010 F. Bayart, <a href="http://www.bibmath.net/dico/index.php3?action=affiche&amp;quoi=./i/indicateureuler.html">Indicateur d'Euler</a>
%H A000010 A. Bogomolny, <a href="http://www.cut-the-knot.org/blue/Euler.shtml">Euler Function and Theorem</a>
%H A000010 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=EulersPhi">Euler's phi function</a>
%H A000010 R. D. Carmichael, <a href="/A002180/a002180.pdf">A table of the values of m corresponding to given values of phi(m)</a>, Amer. J. Math., 30 (1908),394-400. [Annotated scanned copy]
%H A000010 K. Ford, <a href="http://arXiv.org/abs/math.NT/9907204">The number of solutions of phi(x)=m</a>, arXiv:math/9907204 [math.NT], 1999.
%H A000010 Kevin Ford, Florian Luca and Pieter Moree, <a href="http://arxiv.org/abs/1108.3805">Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields</a>, arXiv:1108.3805 [math.NT], 2011.
%H A000010 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1euler.html">The Euler phi function</a>
%H A000010 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000010 E. Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com/2010/07/totient-carnival.html">Totient Carnival partitions</a>, Psychedelic Geometry Blogspot
%H A000010 M. Lal and P. Gillard, <a href="http://dx.doi.org/10.1090/S0025-5718-69-99858-5">Table of Euler's phi function, n < 10^5</a>, Math. Comp., 23 (1969), 682-683.
%H A000010 D. N. Lehmer, <a href="http://projecteuclid.org/euclid.bams/1183425137">Review of Dickson's History of the Theory of Numbers</a>, Bull. Amer. Math. Soc., 26 (1919), 125-132.
%H A000010 Mathforum, <a href="http://mathforum.org/library/drmath/view/51541.html">Proving phi(m) Is Even</a>
%H A000010 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H A000010 Graeme McRae, <a href="http://2000clicks.com/MathHelp/NumberFactorsTotientFunction.aspx">Euler's Totient Function</a>
%H A000010 Carl Pomerance and Hee-Sung Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper7.pdf">Variant of a theorem of Erdos on the sum-of-proper-divisors function</a>, Math. Comp., to appear (2014).
%H A000010 Primefan, <a href="http://primefan.tripod.com/Phi500.html">Euler's Totient Function Values For n=1 to 500, with Divisor Lists</a>
%H A000010 Marko Riedel, <a href="http://www.mathematik.uni-stuttgart.de/~riedelmo/combnumth.html">Combinatorics and number theory page.</a>
%H A000010 K. Schneider, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/EulerPhifunction.html">Euler phi-function</a>
%H A000010 W. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4206.pdf">Euler's Totient Function And The Theorem Of Euler</a>
%H A000010 U. Sondermann, <a href="http://home.earthlink.net/~usondermann/eulertot.html">Euler's Totient Function</a>
%H A000010 W. A. Stein, <a href="http://modular.math.washington.edu/edu/Fall2001/124/lectures/lecture6/html/node3.html">Phi is a Multiplicative Function</a>
%H A000010 G. Villemin, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Nombre/TotEuler.htm">Totient d'Euler</a>
%H A000010 A. de Vries, <a href="http://math-it.org/Mathematik/Zahlentheorie/Zahl/ZahlApplet.html">The prime factors of an integer (along with Euler's phi and Carmichael's lambda functions)</a>
%H A000010 K. W. Wegner, <a href="/A002180/a002180_1.pdf">Values of phi(x) = n for n from 2 through 1978</a>, mimeographed manuscript, no date [Annotated scanned copy]
%H A000010 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/ModuloMultiplicationGroup.html">MathWorld: Modulo Multiplication Group</a>
%H A000010 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/MoebiusTransform.html">MathWorld: Moebius Transform</a>
%H A000010 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/TotientFunction.html">MathWorld: Totient Function</a>
%H A000010 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_phi_function">Euler's totient function</a>
%H A000010 Wikipedia, <a href="http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n">Multiplicative group of integers modulo n</a>
%H A000010 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/EulerPhi/03/02">First 50 values of phi(n)</a>
%H A000010 G. Xiao, Numerical Calculator, <a href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html">To display phi(n) operate on "eulerphi(n)"</a>
%H A000010 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000010 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A000010 phi(n) = n*Product_{distinct primes p dividing n} (1-1/p).
%F A000010 Sum_{ d divides n } phi(d) = n.
%F A000010 phi(n) = Sum_{ d divides n } mu(d)*n/d, i.e., the Moebius transform of the natural numbers; mu() = Moebius function A008683().
%F A000010 Dirichlet generating function sum_{n >= 1} phi(n)/n^s = zeta(s-1)/zeta(s). Also Sum_{n >= 1} phi(n)*x^n/(1-x^n) = x/(1-x)^2.
%F A000010 Multiplicative with a(p^e) = (p-1)*p^(e-1). - _David W. Wilson_, Aug 01 2001
%F A000010 Sum_{n >= 1} [phi(n)*log(1-x^n)/n] = -x/(1-x) for -1 < x < 1 (cf. A002088) - _Henry Bottomley_, Nov 16 2001
%F A000010 a(n) = binomial(n+1, 2) - sum{i = 1, n - 1, a(i)*floor(n/i)} (see A000217 for inverse) - _Jon Perry_, Mar 02 2004
%F A000010 It is a classical result (certainly known to Landau, 1909) that lim inf n/phi(n) = 1 (taking n to be primes), lim sup n/(phi(n) log log n) = e^{gamma}, with gamma = Euler's constant (taking n to be products of consecutive primes starting from 2 and applying Mertens' theorem). See e.g. Ribenboim, pp. 319-320. - Pieter Moree, Sep 10 2004
%F A000010 a(n) = sum_(i = 1..n) | k(n, i)| where k(n, i) is the Kronecker symbol. Also a(n) = #{ 1 <= i <= n : k(n, i) = 0}. - _Benoit Cloitre_, Aug 06 2004
%F A000010 Conjecture : limit Sum((-1)^i/(i * phi(i)) 2 <= i <= Infinity) exists and is ca. 0.558. - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004
%F A000010 From _Enrique Pérez Herrero_, Sep 07 2010: (Start)
%F A000010 a(n) = sum_{i = 1}^{n}{floor(sigma_k(i*n)/sigma_k(i)*sigma_k(n))}, where sigma_2 is A001157.
%F A000010 a(n) = sum_{i = 1}^{n}{floor(tau_k(i*n)/tau_k(i)*tau_k(n))}, where tau_3 is A007425.
%F A000010 a(n) = sum_{i = 1}^{n}{floor(rad(i*n)/rad(i)*rad(n))}, where rad is A007947 (End)
%F A000010 a(n) = A173557(n) * A003557(n). - _R. J. Mathar_, Mar 30 2011
%F A000010 phi(p*n) = phi(n)*[floor(((n+p-1) mod p)/(p-1))+p-1], for prime p. - _Gary Detlefs_, Apr 21 2012
%F A000010 a(n), n odd = 2 * A135303((n-1)/2) * A003558((n-1)/2) or phi(n) = 2 * c * k; the Coach theorem of Pedersen et al. Cf. A135303. - _Gary W. Adamson_, Aug 15 2012
%F A000010 G.f.: sum(n>=1, mu(n)*x^n/(1-x^n)^2), where mu(n)=A008683(n). - _Mamuka Jibladze_, Apr 05 2015
%F A000010 a(n) = n - cototient(n) = n - A051953(n). - _Omar E. Pol_, May 14 2016
%e A000010 G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 6*x^7 + 4*x^8 + 6*x^9 + 4*x^10 + ...
%e A000010 a(8) = 4 with {1, 3, 5, 7} units modulo 8. a(10) = 4 with {1, 3, 7, 9} units modulo 10. - _Michael Somos_, Aug 27 2013
%p A000010 with(numtheory): A000010 := phi; [ seq(phi(n), n=1..100) ]; # version 1
%p A000010 with(numtheory): phi := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := n*mul((1-1/t1[i][1]),i=1..nops(t1)); end; # version 2
%t A000010 Array[EulerPhi, 70]
%o A000010 (Axiom) [eulerPhi(n) for n in 1..100]
%o A000010 (MAGMA) [ EulerPhi(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o A000010 (PARI) {a(n) = if( n==0, 0, eulerphi(n))}; /* _Michael Somos_, Feb 05 2011 */
%o A000010 (Sage)
%o A000010 # euler_phi is a standard function in Sage.
%o A000010 def A000010(n): return euler_phi(n)
%o A000010 def A000010_list(n): return [ euler_phi(i) for i in range(1,n+1)]
%o A000010 # Jaap Spies, Jan 07 2007
%o A000010 (PARI) { for (n=1, 100000, write("b000010.txt", n, " ", eulerphi(n))); } \\ _Harry J. Smith_, Apr 26 2009
%o A000010 (Sage) [euler_phi(n)for n in xrange(1,70)] # _Zerinvary Lajos_, Jun 06 2009
%o A000010 (Maxima) makelist(totient(n),n,0,1000); /* _Emanuele Munarini_, Mar 26 2011 */
%o A000010 (Haskell) a n = length (filter (==1) (map (gcd n) [1..n])) -- _Allan C. Wechsler_, Dec 29 2014
%Y A000010 Cf. A008683, A003434, A007755, A049108, A002202 (values).
%Y A000010 Cf. A005277 (nontotient numbers). For inverse see A002181, A006511, A058277.
%Y A000010 Jordan function J_k(n) is a generalization - see A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
%Y A000010 Cf. A054521, A023022, A054525, A134540.
%Y A000010 Row sums of triangle A134540.
%Y A000010 Equals right and left borders of triangle A159937. - _Gary W. Adamson_, Apr 26 2009
%Y A000010 Row sums of A127448. - _Mats Granvik_, May 28 2008
%Y A000010 Equals row sums of triangle A143239 (a consequence of the Dedekind-Liouville rule, see Concrete Mathematics p. 137). - _Gary W. Adamson_, Aug 01 2008
%Y A000010 Equals row sums of triangle A143353 and of  A143276.
%Y A000010 Cf. A003558, A135303
%K A000010 easy,core,nonn,mult,nice,hear
%O A000010 1,3
%A A000010 _N. J. A. Sloane_


%I A011773
%S A011773 1,1,2,2,4,2,6,4,6,4,10,2,12,6,4,8,16,6,18,4,6,10,22,4,20,12,18,6,28,
%T A011773 4,30,16,10,16,12,6,36,18,12,4,40,6,42,10,12,22,46,8,42,20,16,12,52,
%U A011773 18,20,12,18,28,58,4,60,30,6,32,12,10,66,16,22,12
%N A011773 Variant of Carmichael's lambda function: a(p1^e1*...*pN^eN) = LCM((p1-1)*p1^(e1-1),...,(pN-1)*pN^(eN-1)).
%H A011773 Reinhard Zumkeller, <a href="/A011773/b011773.txt">Table of n, a(n) for n = 1..10000</a>
%H A011773 L. Blum; M. Blum; M. Shub, <a href="http://dx.doi.org/10.1137/0215025">A simple unpredictable pseudorandom number generator</a>, SIAM J. Comput. 15 (1986), no. 2, 364-383. see p. 377.
%H A011773 J.-H. Evertse and E. van Heyst, <a href="http://dx.doi.org/10.1007/3-540-46877-3_8">Which new RSA signatures can be computed from some given RSA signatures?</a>, Proceedings of Eurocrypt'90, Lect. Notes Comput. Sci., 473, Springer-Verlag, pp. 84-97, see page 86.
%H A011773 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelFunction.html">Carmichael Function.</a>
%H A011773 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModuloMultiplicationGroup.html">Modulo Multiplication Group.</a>
%F A011773 a(n) = A002322(2*n), for n != 2. - _Vladeta Jovovic_, Feb 28 2004
%F A011773 a(n) = LCM(A085730(A095874(A027748(n,k)^A124010(n,k))): k=1..A001221(n)). - _Reinhard Zumkeller_, Feb 16 2012
%t A011773 Table[ If[ n==1, 1, LCM@@Map[ (#1[ [ 1 ] ]-1)*#1[ [ 1 ] ]^(#1[ [ 2 ] ]-1)&, FactorInteger[ n ] ] ], {n, 1, 70} ] (* _Olivier Gérard_, Aug 1997 *)
%o A011773 (PARI) a(n)=lcm( apply( f -> (f[1]-1)*f[1]^(f[2]-1), Vec(factor(n)~)))  \\ _M. F. Hasler_, Oct 23 2011
%o A011773 (Haskell)
%o A011773 a011773 n = foldl lcm 1 $ map (a085730 . a095874) $
%o A011773                           zipWith (^) (a027748_row n) (a124010_row n)
%o A011773 -- _Reinhard Zumkeller_, Feb 16 2012
%Y A011773 Cf. A002322.
%K A011773 nonn,nice,easy
%O A011773 1,3
%A A011773 Thierry Moreau (Thierry.Moreau(AT)connotech.com), _Simon Plouffe_
%E A011773 Description corrected by _Antti Karttunen_, Jan 09 2000
%E A011773 Definition made more explicit by _M. F. Hasler_, Oct 23 2011



Even though there are a large number of sequences in the OEIS, at least
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TEST: APPLY VARIOUS TRANSFORMATIONS TO SEQUENCE AND LOOK IT
UP IN THE ENCYCLOPEDIA AGAIN

        SUCCESS
        (limited to 100 matches):

Transformation T006 gave a match with the following sequences (up to a limit of 10):
%I A037225
%S A037225 1,2,4,6,6,10,12,8,16,18,12,22,20,18,28,30,20,24,36,24,40,42,24,46,42,
%T A037225 32,52,40,36,58,60,36,48,66,44,70,72,40,60,78,54,82,64,56,88,72,60,72,
%U A037225 96,60,100,102,48,106,108,72
%N A037225 phi(2n+1).
%C A037225 Bisection of A000010 (cf. A062570).
%D A037225 Brillhart, John; Lomont, J. S.; Morton, Patrick. Cyclotomic properties of the Rudin-Shapiro polynomials. J. Reine Angew. Math.288 (1976), 37--65.  See Table 2. MR0498479 (58 #16589). - From _N. J. A. Sloane_, Jun 06 2012
%D A037225 V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems [in Russian], Problemy Peredachi Informatsii, 43 (No. 3, 2007), 39-53.
%H A037225 T. D. Noe, <a href="/A037225/b037225.txt">Table of n, a(n) for n=0..10000</a>
%Y A037225 Cf. A000010, A062570.
%K A037225 nonn
%O A037225 0,2
%A A037225 _N. J. A. Sloane_.

Transformation T099 gave a match with the following sequences (up to a limit of 10):
%I A002202 M0987 N0371
%S A002202 1,2,4,6,8,10,12,16,18,20,22,24,28,30,32,36,40,42,44,46,48,52,54,56,
%T A002202 58,60,64,66,70,72,78,80,82,84,88,92,96,100,102,104,106,108,110,112,
%U A002202 116,120,126,128,130,132,136,138,140,144,148,150,156,160,162,164,166,168,172,176
%N A002202 Values taken by totient function phi(m) (A000010).
%C A002202 These are the numbers n such that for some m the multiplicative group mod m has order n.
%C A002202 Maier & Pomerance show that there are about x * exp(c (log log log x)^2)/log x members of this sequence up to x, with c = 0.81781465... (A234614); see the paper for details on making this precise. - _Charles R Greathouse IV_, Dec 28 2013
%C A002202 A264739(a(n)) = 1; a(n) occurs A058277(n) times in A007614. - _Reinhard Zumkeller_, Nov 26 2015
%D A002202 J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
%D A002202 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002202 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002202 T. D. Noe, <a href="/A002202/b002202.txt">Table of n, a(n) for n = 1..10000</a>
%H A002202 K. Ford, <a href="http://www.ams.org/era/1998-04-05/S1079-6762-98-00043-2/home.html">The distribution of totients</a>, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
%H A002202 Helmut Maier and Carl Pomerance, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4934.pdf">On the number of distinct values of Euler's phi-function</a>, Acta Arithmetica 49:3 (1988), pp. 263-275.
%H A002202 S. Sivasankaranarayana Pillai, <a href="http://dx.doi.org/10.1090/S0002-9904-1929-04799-2">On some functions connected with phi(n)</a>, Bull. Amer. Math. Soc. 35 (1929), 832-836.
%H A002202 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientValenceFunction.html">Totient Valence Function</a>
%p A002202 with(numtheory); t1 := [seq(nops(invphi(n)), n=1..300)]; t2 := []: for n from 1 to 300 do if t1[n] <> 0 then t2 := [op(t2), n]; fi; od: t2;
%t A002202 phiQ[m_] := Select[Range[m+1, 2m*Product[(1-1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1 ] != {}; Select[Range[176], phiQ] (* _Jean-François Alcover_, May 23 2011, after Maxim Rytin *)
%o A002202 (PARI) lst(lim)=my(P=1,q,v);forprime(p=2,default(primelimit), if(eulerphi(P*=p)>=lim,q=p;break));v=vecsort(vector(P/q*lim\eulerphi(P/q),k,eulerphi(k)),,8);select(n->n<=lim,v) \\ _Charles R Greathouse IV_, Apr 16 2012
%o A002202 (PARI) select(istotient,vector(100,i,i)) \\ _Charles R Greathouse IV_, Dec 28 2012
%o A002202 (Haskell)
%o A002202 import Data.List.Ordered (insertSet)
%o A002202 a002202 n = a002202_list !! (n-1)
%o A002202 a002202_list = f [1..] (tail a002110_list) [] where
%o A002202    f (x:xs) ps'@(p:ps) us
%o A002202      | x < p = f xs ps' $ insertSet (a000010' x) us
%o A002202      | otherwise = vs ++ f xs ps ws
%o A002202      where (vs, ws) = span (<= a000010' x) us
%o A002202 -- _Reinhard Zumkeller_, Nov 22 2015
%Y A002202 Cf. A000010, A002180, A032446, A058277.
%Y A002202 Cf. A002110, A007614, A007617 (complement).
%Y A002202 Cf. A083533 (first differences), A264739.
%K A002202 nonn,nice
%O A002202 1,2
%A A002202 _N. J. A. Sloane_

%I A049445
%S A049445 1,2,4,6,8,10,12,16,18,20,21,24,32,34,36,40,42,48,55,60,64,66,68,69,
%T A049445 72,80,81,84,92,96,108,110,115,116,120,126,128,130,132,136,138,144,
%U A049445 155,156,160,162,168,172,180,184,185,192,204,205,212,216,220,222,228
%N A049445 Numbers n with property that the number of 1's in binary expansion of n (see A000120) divides n.
%C A049445 If instead of base 2 we take base 10, then we have the so-called Harshad or Niven numbers (i.e., positive integers divisible by the sum of their digits; A005349). - _Emeric Deutsch_, Apr 11 2007
%C A049445 A199238(a(n)) = 0. - _Reinhard Zumkeller_, Nov 04 2011
%H A049445 T. D. Noe, <a href="/A049445/b049445.txt">Table of n, a(n) for n=1..1000</a>
%F A049445 {n: A000120(n) | n}. - _R. J. Mathar_, Mar 03 2008
%F A049445 a(n) seems to be asymptotic to c*n*log(n) where 0.7 < c < 0.8. - _Benoit Cloitre_, Jan 22 2003
%F A049445 Heuristically, c should be 1/(2*log(2)), since a random d-bit number should have probability approximately 2/d of being in the sequence. - _Robert Israel_, Aug 22 2014
%F A049445 A049445 = { n: A199238(n)=0 }. - _M. F. Hasler_, Oct 09 2012
%e A049445 20 is in the sequence because 20 is written 10100 in binary and 1 + 1 = 2, which divides 20.
%e A049445 21 is in the sequence because 21 is written 10101 in binary and 1 + 1 + 1 = 3, which divides 21.
%e A049445 22 is not in the sequence because 22 is written 10110 in binary 1 + 1 + 1 = 3, which does not divide 22.
%p A049445 a:=proc(n) local n2: n2:=convert(n,base,2): if n mod add(n2[i],i=1..nops(n2)) = 0 then n else fi end: seq(a(n),n=1..300); # _Emeric Deutsch_, Apr 11 2007
%t A049445 binHarshadQ[n_] := Divisible[n, Count[IntegerDigits[n, 2], 1]]; Select[Range[228], binHarshadQ] (* _Jean-François Alcover_, Dec 01 2011 *)
%t A049445 Select[Range[300],Divisible[#,DigitCount[#,2,1]]&] (* _Harvey P. Dale_, Mar 20 2016 *)
%o A049445 (PARI) for(n=1,1000,b=binary(n):l=length(b); if(n%sum(i=1,l, component(b,i))==0,print1(n,",")))
%o A049445 (PARI) is_A049445(n)={n%norml2(binary(n))==0} \\ _M. F. Hasler_, Oct 09 2012(PARI) isok(n) = ! (n % hammingweight(n)); \\ _Michel Marcus_, Feb 10 2016
%o A049445 (Haskell)
%o A049445 a049445 n = a049445_list !! (n-1)
%o A049445 a049445_list = map (+ 1) $ elemIndices 0 a199238_list
%o A049445 -- _Reinhard Zumkeller_, Nov 04 2011
%o A049445 (Python)
%o A049445 A049445 = [n for n in range(1,10**5) if not n % sum([int(d) for d in bin(n)[2:]])] # _Chai Wah Wu_, Aug 22 2014
%Y A049445 Cf. A000120, A005349, A199238.
%K A049445 nonn,easy,nice,base
%O A049445 1,2
%A A049445 _N. J. A. Sloane_
%E A049445 More terms from _Michael Somos_
%E A049445 Edited by _N. J. A. Sloane_, Oct 07 2005 and May 16 2008

%I A177917
%S A177917 1,2,4,6,8,10,12,16,18,20,24,30,36,40,42,48,58,60,72,78,80,84,90,110,
%T A177917 114,116,120,126,144,156,168,174,180,210,220,222,228,232,234,240,252,
%U A177917 290,312,330,336,342,348,360,390,420,440,444,456,464,468,504,522,546
%N A177917 Numbers n such that n^3 divides 17^(n^2)-1.
%C A177917 The first 9 terms (and others) of A218087 are in this sequence A177917 as well as in A128397. These two sequences have many terms in common, but A177917 \ A128397 = {10, 30, 58, 90, 110,...} and A128397 \ A177917 = {580, 624, 660, 684, 696, 720,...}. - _M. F. Hasler_, Oct 22 2012
%t A177917 Select[Range[600],Divisible[17^#^2-1,#^3]&] (* _Harvey P. Dale_, Jul 19 2015 *)
%o A177917 (PARI) is_A177917(n)=Mod(17,n^3)^(n^2)==1  \\ - _M. F. Hasler_, Oct 21 2012
%Y A177917 Cf. A129211, A129212, A177905, A177907, A127102, A177909, A177243, A177911, A177912, A177913, A177914, A177915, A177916, A177918, A177919, A177920.
%Y A177917 Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, ...,  A128397.
%K A177917 nonn
%O A177917 1,2
%A A177917 _Alexander Adamchuk_, May 14 2010

%I A085150
%S A085150 0,1,2,4,6,8,10,12,16,18,20,22,28,40,42,44,52,66,68,78,80,92,100,102,
%T A085150 106,210,214,232,534,676,772,822,964,1020,1184,1498,2304,2348,7738,
%U A085150 9488,11250,12760,12798,25336,27728,35242,41730,46576,55458,73908,94412,99088
%N A085150 Numbers n such that n!!!!!!+1 is prime.
%C A085150 The search for multifactorial primes started by Ray Ballinger
%C A085150 is now continued by a team of volunteers on the website of Ken Davis (see link).
%H A085150 Ken Davis, <a href="http://mfprimes.free-dc.org/mfdata/">Status of Search for Multifactorial Primes.</a>
%H A085150 Ken Davis, <a href="http://mfprimes.free-dc.org/mfdata/f06p.html"> Results for n!6+1.</a>
%Y A085150 Cf. A085158 (sextuple factorials), A051592.
%K A085150 hard,nonn
%O A085150 1,3
%A A085150 _Hugo Pfoertner_, Jun 23 2003
%E A085150 More terms from _Hugo Pfoertner_, Aug 17 2004
%E A085150 Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

%I A002174 M0986 N0370
%S A002174 1,2,4,6,8,10,12,16,18,20,22,24,28,30,32,36,40,42,44,46,48,52,54,56,
%T A002174 58,60,64,66,70,72,78,80,82,84,88,90,92,96,100,102,104,106,108,110,
%U A002174 112,116,120,126,128,130,132,136,138,140,144,148,150,156,160,162,164,166,168
%N A002174 Values taken by reduced totient function psi(n).
%C A002174 If p is a Sophie Germain prime (A005384), then 2p is here. - _T. D. Noe_, Aug 13 2008
%C A002174 Terms of A002322, sorted and multiple values taken just once. - _Vladimir Joseph Stephan Orlovsky_, Jul 21 2009
%D A002174 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002174 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002174 T. D. Noe, <a href="/A002174/b002174.txt">Table of n, a(n) for n = 1..10000</a>
%H A002174 D. H. Lehmer, <a href="http://dx.doi.org/10.17226/18678">Guide to Tables in the Theory of Numbers</a>, Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
%H A002174 Florian Luca and Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/rangeoflambda13.pdf">On the range of Carmichael's universal-exponent function</a>, Acta Arithmetica 162 (2014), pp.289-308.
%H A002174 C. Moreau, <a href="http://www.numdam.org/item?id=NAM_1898_3_17__293_0">Sur quelques théorèmes d'arithmétique</a>, Nouvelles Annales de Mathématiques, 17 (1898), 293-307.
%F A002174 n (log n)^0.086 << a(n) << n (log n)^0.36 where << is the Vinogradov symbol, see Luca & Pomerance. - _Charles R Greathouse IV_, Dec 28 2013
%t A002174 lst={}; Do[AppendTo[lst, CarmichaelLambda[n]], {n, 6*7!}]; lst; Take[Union[lst], 123] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2009 *)
%t A002174 (* warning: there seems to be no guarantee that no terms near the end are omitted! - _Joerg Arndt_, Dec 23 2014 *)
%t A002174 TakeWhile[Union@ Table[CarmichaelLambda@ n, {n, 10^6}], # <= 168 &] (* _Michael De Vlieger_, Mar 19 2016 *)
%Y A002174 Cf. A002322, A002396, A143407, A143408.
%K A002174 nonn,nice
%O A002174 1,2
%A A002174 _N. J. A. Sloane_.
%E A002174 More terms from _T. D. Noe_, Aug 13 2008

%I A213708
%S A213708 0,1,2,4,6,8,10,12,16,18,20,24,28,32,34,36,40,44,48,52,56,60,64,66,68,
%T A213708 72,76,80,84,88,92,96,100,104,108,112,116,120,126,128,130,132,136,140,
%U A213708 144,148,152,156,160,164,168,172,176,180,184,190,192,196,200,204,208,212,216,222,226,232,238,244,250,256,258,260,264,268,272,276
%N A213708 a(n) = Least inverse of A071542, i.e. minimal i such that A071542(i)=n.
%C A213708 Also the positions in A071542 where new records appear, record values appearing in the ascending order, i.e. as A001477. (Because A071542 is a monotone and surjective function).
%H A213708 Antti Karttunen, <a href="/A213708/b213708.txt">Table of n, a(n) for n = 0..16500</a>
%o A213708 (Scheme with _Antti Karttunen_'s intseq-additions): (define A213708 (RECORD-POS 0 0 A071542))
%Y A213708 Cf. A173601 for the greatest inverse. A086876 gives the first differences. Cf. A179016, A213730.
%K A213708 nonn
%O A213708 0,3
%A A213708 _Antti Karttunen_, Oct 24 2012

%I A015926
%S A015926 1,2,4,6,8,10,12,16,18,24,30,31,32,36,42,48,64,66,72,78,84,90,96,102,
%T A015926 114,126,138,144,168,174,176,186,192,210,222,234,246,252,258,282,288,
%U A015926 318,336,354,366,390,396,402,426,438,456,474,496,498,504,510,534,546
%N A015926 Positive integers n such that 2^n == 2^6 (mod n).
%C A015926 The odd terms are given by A215610.
%H A015926 T. D. Noe, <a href="/A015926/b015926.txt">Table of n, a(n) for n = 1..1000</a>
%H A015926 OEIS Wiki, <a href="/wiki/2^n mod n">2^n mod n</a>
%t A015926 Select[Range[1000], Mod[2^# - 2^6, #] == 0 &] (* _T. D. Noe_, Aug 17 2012 *)
%Y A015926 Cf. A015919, A015921, A015922, A015924, A015925, A015927, A015929, A015931, A015932, A015935, A015937.
%K A015926 nonn
%O A015926 1,2
%A A015926 _Robert G. Wilson v_
%E A015926 Edited by _Max Alekseyev_, Jul 30 2011

%I A093891
%S A093891 1,2,4,6,8,10,12,16,18,20,24,28,30,32,36,40,42,48,54,56,60,64,66,70,
%T A093891 72,78,80,84,88,90,96,100,104,108,112,120,126,128,132,140,144,150,156,
%U A093891 160,162,168,176,180,192,196,198,200,204,208,210,216,220,224,228,234,240
%N A093891 Numbers n such that every prime up to sigma(n) is a sum of divisors of n.
%C A093891 Sequence is infinite as sigma (2^n)= 2^(n+1)-1 and a(2^n) = pi(2^(n+1)-1).
%C A093891 Does this sequence include any non-members of A005153 other than 10, 70 and 836? - _Franklin T. Adams-Watters_, Apr 28 2006
%C A093891 The answer to the previous comment is yes, this sequence has many terms that are not in A005153. See A174434. [From _T. D. Noe_, Mar 19 2010]
%e A093891 4 is a member as sigma(4) = 7 and all the primes up to 7 are a partial sum of divisors of 4.
%e A093891 Divisors of 4 are 1, 2 and 4 . Primes arising are 2, 3= 1+2, 5 = 1+4 and 7 = 1 + 2 + 4.
%Y A093891 Cf. A093890, A093892.
%Y A093891 Cf. A005153.
%K A093891 nonn
%O A093891 1,2
%A A093891 _Amarnath Murthy_, Apr 23 2004
%E A093891 More terms from _Franklin T. Adams-Watters_, Apr 28 2006

%I A267817
%S A267817 1,2,4,6,8,10,12,16,18,20,24,30,32,36,40,42,48,50,54,60,64,68,72,78,
%T A267817 80,84,90,96,100,108,110,114,120,126,128,136,144,150,156,160,162,168,
%U A267817 180,192,200,204,210,216,220,222,228,234,240
%N A267817 Numbers n with property that n is divisible by A268336(n).
%C A267817 Squarefree terms: 1, 2, 6, 10, 30, 42, 78, 110, 114, 210, 222, ...
%H A267817 Charles R Greathouse IV, <a href="/A267817/b267817.txt">Table of n, a(n) for n = 1..10000</a>
%H A267817 Juri-Stepan Gerasimov, _divisors (k^k mod n) of n_, SeqFan list, Feb, 13, 2016
%e A267817 10 is in this sequence because 10/A268336(10) = 10/2 = 5.
%o A267817 (PARI) is(n)=my(f=factor(n)[, 1],m=n); for(k=1, #f, m=lcm(m, f[k]-1)); m/=n; m && n%m==0 \\ _Charles R Greathouse IV_, Feb 22 2016
%Y A267817 Cf. A174824, A268336.
%K A267817 nonn
%O A267817 1,2
%A A267817 _Juri-Stepan Gerasimov_, Feb 13 2016
%E A267817 a(16) inserted by _Charles R Greathouse IV_, Feb 22 2016

%I A097380
%S A097380 1,2,4,6,8,10,12,16,18,22,24,28,30,32,36,42,46,48,52,54,56,58,60,64,
%T A097380 66,70,72,78,82,96,100,102,104,106,108,112,120,126,128,130,138,144,
%U A097380 148,150,156,162,166,172,178,180,190,192,196,198,200,208,210,216,222,224,226
%N A097380 Numbers m such that 1+CubeFreeKernel(m) is prime.
%C A097380 A097377(a(n)) = A007948(a(n))+1 is prime.
%e A097380 m=216=(2*3)^3 -> A097377(216) = 1+(2*3)^2 = 37 = A000040(12), therefore 216 is a term.
%Y A097380 Cf. A089189, A097379, A097381.
%K A097380 nonn
%O A097380 1,2
%A A097380 _Reinhard Zumkeller_, Aug 11 2004


List of transformations used:
T006  elements of odd index in the sequence
T099  sort terms, remove duplicates

Abbreviations used in the above list of transformations:
u[j]    =       j-th term of the sequence
v[j]    =       u[j]/(j-1)!
Sn(z)   =       ordinary generating function
En(z)   =       exponential generating function



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