OFFSET
1,1
COMMENTS
First 13 terms are all terms up to 5 * 10^14.
Each term is 72 modulo 144.
Let g(k) (= A000001(n)) denote the number of groups with k elements.
The factorization of even k where g(k) = c for small c is very constrained.
This is a subsequence of A392129 (length 6 sequences), which has more comments.
If we write k = 144m + 72, then k+1 = 144m + 73, k+2 = 2(72m + 37), k+4 = 4(36m + 19), and k+6 = 6(24m + 13).
The only way g(k+2) can be 2 is if 72m + 37 is prime (Olsson 2006).
The only way g(k+4) can be 4 is if 36m + 19 is prime (Miller 1932).
The only way g(k+6) can be 6 is if 24m + 13 is prime (Mahmoud 2024).
For g(k+1) to be 1, 144m + 73 must be squarefree (necessary and easy to sieve, but not sufficient).
This sequence is infinite assuming the Bateman-Horn conjecture is true. We can construct a subsequence where the odd k+i also have a nice factorization by restricting k+3 mod 5^2, then k+5 mod 7^4, then k+7 mod 11^4*43^2 (or many other examples). E.g. if k = 233992702112400m + 1301555283912, with all of 233992702112400m + 1301555283913, 116996351056200m + 650777641957, 15599513474160m + 86770352261, 58498175528100m + 325388820979, 682194466800m + 3794621819, 38998783685400m + 216925880653, and 4088422800m + 22741343 prime, then that's a subsequence of this sequence which is infinite assuming Bateman-Horn (these large linear forms show up when we factor out the gcd from k+1, k+2, etc., and the gcds are constructed so that they have the requisite group number, and then we multiply them by a prime that does not interact).
LINKS
Keith Conrad, Groups of order p^3.
John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Gabriel Eiseman, source code to compute this sequence, 2025.
Robin Jones, Does the sequence 1,2,3,4,5,6 appear in the number of groups of order n up to isomorphism?, Math StackExchange.
Aban S. Mahmoud, Orders for which there exist exactly six or seven groups, arXiv:2405.04794 [math.GR], 2024.
G. A. Miller, Orders for which There Exist Exactly Four or Five Groups, Proceedings of the National Academy of Sciences of the United States of America, vol. 18, no. 7, 1932, pp. 511-514.
Jørn B. Olsson, Three-group numbers, 2006.
EXAMPLE
For k = 5973822114120, the factorizations of k+1 thru k+7 are as follows:
5973822114121 = 37 * 109 * 487 * 3041551 (squarefree with no relations p | q - 1, so we have one group per Olsson 2006, namely the cyclic group)
5973822114122 = 2 * 2986911057061 (squarefree with one relation because 2 | 2986911057061 - 1, so we have two groups per Olsson 2006, namely the cyclic group C2p and the dihedral group D2p)
5973822114123 = 3 * 19 * 137 * 764991947 (squarefree with 3 | 19 - 1 and 19 | 764991947 and no other relations, so we have three groups per Olsson 2006)
5973822114124 = 2^2 * 1493455528531 (squarefree besides 2^2, with one relation because 2 | 1493455528531 - 1, so we have four groups per Miller 1932, namely C2 x C2 x Cp, C4 x Cp, C2 x D2p, and D4p)
5973822114125 = 5^3 * 2719 * 17576527 (squarefree besides 5^3, no relations, so we have five groups per Miller 1932, namely C5 x C5 x C5 x C47790576913, C5 x C25 x C47790576913, C125 x C47790576913, (C5 semidirect product C25) x C47790576913, and Heisenburg(Z_5) x C47790576913. The five possibilities for the size 125 subgroup are listed in Conrad, and because 47790576913 is coprime to the size of the automorphism group for each of these, there are no nontrivial semidirect products, just the direct product)
5973822114126 = 2 * 3 * 995637019021 (squarefree with 2 | 3 - 1, 2 | 995637019021 - 1, and 3 | 995637019021 - 1, so we have six groups per Mahmoud 2024)
5973822114127 = 7^2 * 97^2 * 17 * 181 * 4211 (squarefree besides 7^2 and 97^2 with no relations p | q - 1, 7 | 97 + 1, and no other relations p | 7 + 1 or p | 97 + 1, so we have seven groups per Mahmoud 2024)
PROG
(Rust) // See Links
CROSSREFS
Cf. A000001 (groups of order n).
Numbers m such that there are precisely k groups of order m: A003277 (k=1), A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7).
KEYWORD
nonn,more
AUTHOR
Gabriel Eiseman, Dec 31 2025
STATUS
approved