A232106 Number of groups of order prime(n)^6.

267, 504, 684, 860, 1192, 1476, 1944, 2264, 2876, 4068, 4540, 6012, 7064, 7664, 8852, 10908, 13136, 14012, 16520, 18292, 19296, 22244, 24296, 27648, 32472, 34964, 36284, 38912, 40356, 43128, 53780, 56992, 62064, 63824, 72828, 74740, 80532, 86504, 90572, 96948

Offset

1,1

Comments

Isomorphism types of groups and nilpotent Lie rings with order prime(n)^6.

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra, 278 (2004), 383-401.

Index entries for sequences related to groups

Formula

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).

Maple

a:= n-> `if`(n<3, [267, 504][n], (c-> 386 +(45 +3*c)*c+
    24*igcd(c, 3) +11*igcd(c, 4) +2*igcd(c, 5))(ithprime(n)-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Nov 17 2017

Mathematica

Table[FiniteGroupCount[Prime[n]^6], {n, 40}] (* Michael De Vlieger, Apr 12 2016 *)

Prog

(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)
(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
(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

Crossrefs

Cf. A000001, A000679, A090091, A090130, A090140, A128604, A232105, A232107.

Cf. A030516.

Sequence in context: A278307 A060402 A306120 * A049014 A186056 A236740

Adjacent sequences: A232103 A232104 A232105 * A232107 A232108 A232109

Keyword

nonn

Author

Eric M. Schmidt, Nov 21 2013

Status

approved