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# dario2994/generate_hcn.py

Last active January 22, 2024 15:54
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Highly composite numbers list
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 #!/usr/bin/env python3 # This program prints all hcn (highly composite numbers) <= MAXN (=10**18) # # The value of MAXN can be changed arbitrarily. When MAXN = 10**100, the # program needs less than one second to generate the list of hcn. from math import log MAXN = 10**18 # Generates a list of the first primes (with product > MAXN). def gen_primes(): primes = [] primes_product = 1 for n in range(2, 10**10): is_prime = True for i in range(2, n): if n % i == 0: is_prime = False if is_prime: primes.append(n) primes_product *= n if primes_product > MAXN: break return primes primes = gen_primes() # Generates a list of the hcn <= MAXN. def gen_hcn(): # List of (number, number of divisors, exponents of the factorization) hcn = [(1, 1, [])] for i in range(len(primes)): new_hcn = [] for el in hcn: new_hcn.append(el) if len(el[2]) < i: continue e_max = el[2][i-1] if i >= 1 else int(log(MAXN, 2)) n = el[0] for e in range(1, e_max+1): n *= primes[i] if n > MAXN: break div = el[1] * (e+1) exponents = el[2] + [e] new_hcn.append((n, div, exponents)) new_hcn.sort() hcn = [(1, 1, [])] for el in new_hcn: if el[1] > hcn[-1][1]: hcn.append(el) return hcn hcn = gen_hcn() # From here on is only pretty printing. print("Number of highly composite numbers less than", MAXN, "is", len(hcn),"\n") def PrintWithCorrectSpaces(a, b, c): aspace = int(log(MAXN, 10)) + 5 bspace = int(log(hcn[-1][1], 10)) + 5 assert(len(a) < aspace) assert(len(b) < bspace) print(a, " "*(aspace-len(a)), b, " "*(bspace-len(b)), c) PrintWithCorrectSpaces("number", "divisors", "factorization") for el in hcn: factorization = "*".join([str(primes[i])+("^"+str(el[2][i]) if el[2][i]>1 else "") for i in range(len(el[2]))]) PrintWithCorrectSpaces(str(el[0]), str(el[1]), factorization)
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 Number of highly composite numbers less than 1000000000000000000 is 156 number divisors factorization 1 1 2 2 2 4 3 2^2 6 4 2*3 12 6 2^2*3 24 8 2^3*3 36 9 2^2*3^2 48 10 2^4*3 60 12 2^2*3*5 120 16 2^3*3*5 180 18 2^2*3^2*5 240 20 2^4*3*5 360 24 2^3*3^2*5 720 30 2^4*3^2*5 840 32 2^3*3*5*7 1260 36 2^2*3^2*5*7 1680 40 2^4*3*5*7 2520 48 2^3*3^2*5*7 5040 60 2^4*3^2*5*7 7560 64 2^3*3^3*5*7 10080 72 2^5*3^2*5*7 15120 80 2^4*3^3*5*7 20160 84 2^6*3^2*5*7 25200 90 2^4*3^2*5^2*7 27720 96 2^3*3^2*5*7*11 45360 100 2^4*3^4*5*7 50400 108 2^5*3^2*5^2*7 55440 120 2^4*3^2*5*7*11 83160 128 2^3*3^3*5*7*11 110880 144 2^5*3^2*5*7*11 166320 160 2^4*3^3*5*7*11 221760 168 2^6*3^2*5*7*11 277200 180 2^4*3^2*5^2*7*11 332640 192 2^5*3^3*5*7*11 498960 200 2^4*3^4*5*7*11 554400 216 2^5*3^2*5^2*7*11 665280 224 2^6*3^3*5*7*11 720720 240 2^4*3^2*5*7*11*13 1081080 256 2^3*3^3*5*7*11*13 1441440 288 2^5*3^2*5*7*11*13 2162160 320 2^4*3^3*5*7*11*13 2882880 336 2^6*3^2*5*7*11*13 3603600 360 2^4*3^2*5^2*7*11*13 4324320 384 2^5*3^3*5*7*11*13 6486480 400 2^4*3^4*5*7*11*13 7207200 432 2^5*3^2*5^2*7*11*13 8648640 448 2^6*3^3*5*7*11*13 10810800 480 2^4*3^3*5^2*7*11*13 14414400 504 2^6*3^2*5^2*7*11*13 17297280 512 2^7*3^3*5*7*11*13 21621600 576 2^5*3^3*5^2*7*11*13 32432400 600 2^4*3^4*5^2*7*11*13 36756720 640 2^4*3^3*5*7*11*13*17 43243200 672 2^6*3^3*5^2*7*11*13 61261200 720 2^4*3^2*5^2*7*11*13*17 73513440 768 2^5*3^3*5*7*11*13*17 110270160 800 2^4*3^4*5*7*11*13*17 122522400 864 2^5*3^2*5^2*7*11*13*17 147026880 896 2^6*3^3*5*7*11*13*17 183783600 960 2^4*3^3*5^2*7*11*13*17 245044800 1008 2^6*3^2*5^2*7*11*13*17 294053760 1024 2^7*3^3*5*7*11*13*17 367567200 1152 2^5*3^3*5^2*7*11*13*17 551350800 1200 2^4*3^4*5^2*7*11*13*17 698377680 1280 2^4*3^3*5*7*11*13*17*19 735134400 1344 2^6*3^3*5^2*7*11*13*17 1102701600 1440 2^5*3^4*5^2*7*11*13*17 1396755360 1536 2^5*3^3*5*7*11*13*17*19 2095133040 1600 2^4*3^4*5*7*11*13*17*19 2205403200 1680 2^6*3^4*5^2*7*11*13*17 2327925600 1728 2^5*3^2*5^2*7*11*13*17*19 2793510720 1792 2^6*3^3*5*7*11*13*17*19 3491888400 1920 2^4*3^3*5^2*7*11*13*17*19 4655851200 2016 2^6*3^2*5^2*7*11*13*17*19 5587021440 2048 2^7*3^3*5*7*11*13*17*19 6983776800 2304 2^5*3^3*5^2*7*11*13*17*19 10475665200 2400 2^4*3^4*5^2*7*11*13*17*19 13967553600 2688 2^6*3^3*5^2*7*11*13*17*19 20951330400 2880 2^5*3^4*5^2*7*11*13*17*19 27935107200 3072 2^7*3^3*5^2*7*11*13*17*19 41902660800 3360 2^6*3^4*5^2*7*11*13*17*19 48886437600 3456 2^5*3^3*5^2*7^2*11*13*17*19 64250746560 3584 2^6*3^3*5*7*11*13*17*19*23 73329656400 3600 2^4*3^4*5^2*7^2*11*13*17*19 80313433200 3840 2^4*3^3*5^2*7*11*13*17*19*23 97772875200 4032 2^6*3^3*5^2*7^2*11*13*17*19 128501493120 4096 2^7*3^3*5*7*11*13*17*19*23 146659312800 4320 2^5*3^4*5^2*7^2*11*13*17*19 160626866400 4608 2^5*3^3*5^2*7*11*13*17*19*23 240940299600 4800 2^4*3^4*5^2*7*11*13*17*19*23 293318625600 5040 2^6*3^4*5^2*7^2*11*13*17*19 321253732800 5376 2^6*3^3*5^2*7*11*13*17*19*23 481880599200 5760 2^5*3^4*5^2*7*11*13*17*19*23 642507465600 6144 2^7*3^3*5^2*7*11*13*17*19*23 963761198400 6720 2^6*3^4*5^2*7*11*13*17*19*23 1124388064800 6912 2^5*3^3*5^2*7^2*11*13*17*19*23 1606268664000 7168 2^6*3^3*5^3*7*11*13*17*19*23 1686582097200 7200 2^4*3^4*5^2*7^2*11*13*17*19*23 1927522396800 7680 2^7*3^4*5^2*7*11*13*17*19*23 2248776129600 8064 2^6*3^3*5^2*7^2*11*13*17*19*23 3212537328000 8192 2^7*3^3*5^3*7*11*13*17*19*23 3373164194400 8640 2^5*3^4*5^2*7^2*11*13*17*19*23 4497552259200 9216 2^7*3^3*5^2*7^2*11*13*17*19*23 6746328388800 10080 2^6*3^4*5^2*7^2*11*13*17*19*23 8995104518400 10368 2^8*3^3*5^2*7^2*11*13*17*19*23 9316358251200 10752 2^6*3^3*5^2*7*11*13*17*19*23*29 13492656777600 11520 2^7*3^4*5^2*7^2*11*13*17*19*23 18632716502400 12288 2^7*3^3*5^2*7*11*13*17*19*23*29 26985313555200 12960 2^8*3^4*5^2*7^2*11*13*17*19*23 27949074753600 13440 2^6*3^4*5^2*7*11*13*17*19*23*29 32607253879200 13824 2^5*3^3*5^2*7^2*11*13*17*19*23*29 46581791256000 14336 2^6*3^3*5^3*7*11*13*17*19*23*29 48910880818800 14400 2^4*3^4*5^2*7^2*11*13*17*19*23*29 55898149507200 15360 2^7*3^4*5^2*7*11*13*17*19*23*29 65214507758400 16128 2^6*3^3*5^2*7^2*11*13*17*19*23*29 93163582512000 16384 2^7*3^3*5^3*7*11*13*17*19*23*29 97821761637600 17280 2^5*3^4*5^2*7^2*11*13*17*19*23*29 130429015516800 18432 2^7*3^3*5^2*7^2*11*13*17*19*23*29 195643523275200 20160 2^6*3^4*5^2*7^2*11*13*17*19*23*29 260858031033600 20736 2^8*3^3*5^2*7^2*11*13*17*19*23*29 288807105787200 21504 2^6*3^3*5^2*7*11*13*17*19*23*29*31 391287046550400 23040 2^7*3^4*5^2*7^2*11*13*17*19*23*29 577614211574400 24576 2^7*3^3*5^2*7*11*13*17*19*23*29*31 782574093100800 25920 2^8*3^4*5^2*7^2*11*13*17*19*23*29 866421317361600 26880 2^6*3^4*5^2*7*11*13*17*19*23*29*31 1010824870255200 27648 2^5*3^3*5^2*7^2*11*13*17*19*23*29*31 1444035528936000 28672 2^6*3^3*5^3*7*11*13*17*19*23*29*31 1516237305382800 28800 2^4*3^4*5^2*7^2*11*13*17*19*23*29*31 1732842634723200 30720 2^7*3^4*5^2*7*11*13*17*19*23*29*31 2021649740510400 32256 2^6*3^3*5^2*7^2*11*13*17*19*23*29*31 2888071057872000 32768 2^7*3^3*5^3*7*11*13*17*19*23*29*31 3032474610765600 34560 2^5*3^4*5^2*7^2*11*13*17*19*23*29*31 4043299481020800 36864 2^7*3^3*5^2*7^2*11*13*17*19*23*29*31 6064949221531200 40320 2^6*3^4*5^2*7^2*11*13*17*19*23*29*31 8086598962041600 41472 2^8*3^3*5^2*7^2*11*13*17*19*23*29*31 10108248702552000 43008 2^6*3^3*5^3*7^2*11*13*17*19*23*29*31 12129898443062400 46080 2^7*3^4*5^2*7^2*11*13*17*19*23*29*31 18194847664593600 48384 2^6*3^5*5^2*7^2*11*13*17*19*23*29*31 20216497405104000 49152 2^7*3^3*5^3*7^2*11*13*17*19*23*29*31 24259796886124800 51840 2^8*3^4*5^2*7^2*11*13*17*19*23*29*31 30324746107656000 53760 2^6*3^4*5^3*7^2*11*13*17*19*23*29*31 36389695329187200 55296 2^7*3^5*5^2*7^2*11*13*17*19*23*29*31 48519593772249600 57600 2^9*3^4*5^2*7^2*11*13*17*19*23*29*31 60649492215312000 61440 2^7*3^4*5^3*7^2*11*13*17*19*23*29*31 72779390658374400 62208 2^8*3^5*5^2*7^2*11*13*17*19*23*29*31 74801040398884800 64512 2^6*3^3*5^2*7^2*11*13*17*19*23*29*31*37 106858629141264000 65536 2^7*3^3*5^3*7*11*13*17*19*23*29*31*37 112201560598327200 69120 2^5*3^4*5^2*7^2*11*13*17*19*23*29*31*37 149602080797769600 73728 2^7*3^3*5^2*7^2*11*13*17*19*23*29*31*37 224403121196654400 80640 2^6*3^4*5^2*7^2*11*13*17*19*23*29*31*37 299204161595539200 82944 2^8*3^3*5^2*7^2*11*13*17*19*23*29*31*37 374005201994424000 86016 2^6*3^3*5^3*7^2*11*13*17*19*23*29*31*37 448806242393308800 92160 2^7*3^4*5^2*7^2*11*13*17*19*23*29*31*37 673209363589963200 96768 2^6*3^5*5^2*7^2*11*13*17*19*23*29*31*37 748010403988848000 98304 2^7*3^3*5^3*7^2*11*13*17*19*23*29*31*37 897612484786617600 103680 2^8*3^4*5^2*7^2*11*13*17*19*23*29*31*37

### Thatyougoon commented Dec 20, 2019

Awesome needed this for a project of mine. Many thanks!

nice

### DarshilShah-12 commented Aug 8, 2021

This was very helpful. Thanks a lot!

### BullHunter3 commented Oct 11, 2021

This has helped me a lot with investigating algorithms to generate highly composite numbers.

I suggest making two changes to further optimize this script:

1. Move generating primes from line 28 to just before def gen_hcn and removing line 68, primes = gen_primes() as it then is unneeded.
2. Move line 44, if n > MAXN: break, to after line 41 to eliminate having to unnecessarily calculate div and appending exponents.

FYI:
Your method is about 10 times faster than the Saino and Saino algorithm (implemented in Python). Their paper can be found at: http://wwwhomes.uni-bielefeld.de/achim/julianmanuscript3.pdf

### EntityPlantt commented Aug 1, 2023

Thank you very much! 🤗 It helped me a lot to decide whether my idea of dynamic programming fits for the problem Six (EJOI 2017).
A little bit sad that it is in only one language, may write it in C++ or JS later

### ivanovmp commented Jan 21, 2024

I rewrote it in C++. Impressive, it is really fast!

https://godbolt.org/z/hxavf6sP9

### BullHunter3 commented Jan 22, 2024

Neat! I may have to revisit the Highly Composite Number problem space.