A271811 Number of non-abelian groups of order prime(n)^6.

256, 493, 673, 849, 1181, 1465, 1933, 2253, 2865, 4057, 4529, 6001, 7053, 7653, 8841, 10897, 13125, 14001, 16509, 18281, 19285, 22233, 24285, 27637, 32461, 34953, 36273, 38901, 40345, 43117, 53769, 56981, 62053, 63813, 72817, 74729, 80521, 86493, 90561, 96937, 103485, 105801, 117281

Offset: 1

Author

Altug Alkan, Apr 14 2016

Keywords

• nonn

Comments

A000688(p^6) is 11 for all prime p.

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..1229
• 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

Crossrefs

Cf. A000688, A030516, A060689, A232106, A271664.

Programs

GAP

A271811 := Concatenation([256, 493], List(Filtered([5..10^4], IsPrime), p -> 3 * p^2 + 39 * p + 333 + 24 * Gcd(p-1, 3) + 11 * Gcd(p-1, 4) + 2 * Gcd(p-1,5))); # Muniru A Asiru, Nov 18 2017

Mathematica

Table[FiniteGroupCount[#] - FiniteAbelianGroupCount[#] &[Prime[n]^6], {n, 43}] (* Michael De Vlieger, Apr 15 2016, after Vladimir Joseph Stephan Orlovsky at A060689 *)

PARI

a(n) = if (n==1, 256, if (n==2, 493, my(p=prime(n)); 3*p^2 + 39*p + 333 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5)));

Formula

a(n) = A232106(n) - 11.

a(n) = A060689(prime(n)^6) = A060689(A030516(n)).

For a prime p > 3, the number of non-abelian groups of order p^6 is 3p^2 + 39p + 333 + 24 gcd(p - 1, 3) + 11 gcd(p - 1, 4) + 2 gcd(p - 1, 5).