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%I #35 Nov 17 2017 20:50:18
%S 267,504,684,860,1192,1476,1944,2264,2876,4068,4540,6012,7064,7664,
%T 8852,10908,13136,14012,16520,18292,19296,22244,24296,27648,32472,
%U 34964,36284,38912,40356,43128,53780,56992,62064,63824,72828,74740,80532,86504,90572,96948
%N Number of groups of order prime(n)^6.
%C Isomorphism types of groups and nilpotent Lie rings with order prime(n)^6.
%H Eric M. Schmidt, <a href="/A232106/b232106.txt">Table of n, a(n) for n = 1..1000</a>
%H M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.11.012">Groups and nilpotent Lie rings whose order is the sixth power of a prime</a>, J. Algebra, 278 (2004), 383-401.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F For a prime p > 3, the number of groups of order p^6 is 3p^2 + 39p + 344 + 24 gcd(p - 1, 3) + 11 gcd(p - 1, 4) + 2 gcd(p - 1, 5).
%p a:= n-> `if`(n<3, [267, 504][n], (c-> 386 +(45 +3*c)*c+
%p 24*igcd(c, 3) +11*igcd(c, 4) +2*igcd(c, 5))(ithprime(n)-1)):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Nov 17 2017
%t Table[FiniteGroupCount[Prime[n]^6], {n, 40}] (* _Michael De Vlieger_, Apr 12 2016 *)
%o (Sage) def A232106(n) : p = nth_prime(n); return 267 if p==2 else 504 if p==3 else 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5)
%o (PARI) a(n) = if(n==1, 267, if (n==2, 504, my(p=prime(n)); 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5))); \\ _Altug Alkan_, Apr 12 2016
%o (GAP) A232106 := Concatenation([267, 504], List(Filtered([5..10^5], IsPrime), p -> 3 * p^2 + 39 * p + 344 + 24 * Gcd(p-1, 3) + 11 * Gcd(p-1, 4) + 2 * Gcd(p-1, 5))); # _Muniru A Asiru_, Nov 16 2017
%Y Cf. A000001, A000679, A090091, A090130, A090140, A128604, A232105, A232107.
%Y Cf. A030516.
%K nonn
%O 1,1
%A _Eric M. Schmidt_, Nov 21 2013
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