A232107 Number of groups of order prime(n)^7.

2328, 9310, 34297, 113147, 750735, 1600573, 5546909, 9380741, 23316851, 71271069, 98488755, 233043067, 384847485, 485930975, 751588475, 1356370173, 2299880351, 2710679045, 4306310927, 5734323819, 6578172579, 9721485395, 12413061671, 17537591045, 26866372821

Offset: 1

Author

Eric M. Schmidt, Nov 21 2013

Keywords

• nonn

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..1000
• E. A. O'Brien and M. R. Vaughan-Lee, The groups of order p^7 for odd prime p, J. Algebra 292, 243-258, 2005.
• Index entries for sequences related to groups

Crossrefs

Cf. A000001, A000679, A090091, A090130, A090140, A232105, A232106.

Cf. A092759.

Programs

GAP

A232107 := Concatenation([2328, 9310, 34297], List(Filtered([7..10^5], IsPrime), p -> 3 * p^5 + 12 * p^4 + 44 * p^3 + 170 * p^2 + 707 * p + 2455 + (4 * p^2 + 44 * p + 291) * Gcd(p-1, 3) + (p^2 + 19 * p + 135) * Gcd(p-1, 4) + (3 * p + 31) * Gcd(p-1, 5) + 4 *  Gcd(p-1, 7) + 5 * Gcd(p-1, 8) +  Gcd(p-1, 9))); # Muniru A Asiru, Nov 16 2017

Maple

a:= n-> `if`(n<4, [2328, 9310, 34297][n], (c-> 3391 +(1242+
    (404 +(122 +(27 +3*c)*c)*c)*c)*c +(339 +(52 +4*c)*c)*igcd(c, 3)+
    (155 +(21 +c)*c)*igcd(c, 4) +(34 +3*c)*igcd(c, 5) +4*igcd(c, 7)+
     5*igcd(c, 8) +igcd(c, 9))(ithprime(n)-1)):
seq(a(n), n=1..25);  # Alois P. Heinz, Nov 17 2017

Sage

def A232107(n) : p = nth_prime(n); return 2328 if p==2 else 9310 if p==3 else 34297 if p==5 else 3*p^5 + 12*p^4 + 44*p^3 + 170*p^2 + 707*p + 2455 + (4*p^2 + 44*p + 291)*gcd(p - 1, 3) + (p^2 + 19*p + 135)*gcd(p - 1, 4) + (3*p + 31)*gcd(p - 1, 5) + 4*gcd(p - 1, 7) + 5*gcd(p - 1, 8) + gcd(p - 1, 9)

Formula

For a prime p > 5, the number of groups of order p^7 is 3p^5 + 12p^4 + 44p^3 + 170p^2 + 707p + 2455 + (4p^2 + 44p + 291)gcd(p - 1, 3) + (p^2 + 19p + 135)gcd(p - 1, 4) + (3p + 31)gcd(p - 1, 5) + 4 gcd(p - 1, 7) + 5 gcd(p - 1, 8) + gcd(p - 1, 9).