|
%I #18 Nov 17 2017 21:12:07
%S 2328,9310,34297,113147,750735,1600573,5546909,9380741,23316851,
%T 71271069,98488755,233043067,384847485,485930975,751588475,1356370173,
%U 2299880351,2710679045,4306310927,5734323819,6578172579,9721485395,12413061671,17537591045,26866372821
%N Number of groups of order prime(n)^7.
%H Eric M. Schmidt, <a href="/A232107/b232107.txt">Table of n, a(n) for n = 1..1000</a>
%H E. A. O'Brien and M. R. Vaughan-Lee, <a href="http://dx.doi.org/10.1016/j.jalgebra.2005.01.019">The groups of order p^7 for odd prime p</a>, J. Algebra 292, 243-258, 2005.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F 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).
%p a:= n-> `if`(n<4, [2328, 9310, 34297][n], (c-> 3391 +(1242+
%p (404 +(122 +(27 +3*c)*c)*c)*c)*c +(339 +(52 +4*c)*c)*igcd(c, 3)+
%p (155 +(21 +c)*c)*igcd(c, 4) +(34 +3*c)*igcd(c, 5) +4*igcd(c, 7)+
%p 5*igcd(c, 8) +igcd(c, 9))(ithprime(n)-1)):
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Nov 17 2017
%o (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)
%o (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
%Y Cf. A000001, A000679, A090091, A090130, A090140, A232105, A232106.
%Y Cf. A092759.
%K nonn
%O 1,1
%A _Eric M. Schmidt_, Nov 21 2013
|
