cp's OEIS Frontend

A359421 a(n) = number of abelian groups of order p^2 - 1, where p = prime(n).

%I A359421 #22 Feb 03 2025 09:37:07
%S A359421 1,3,3,5,3,3,14,6,5,3,11,6,5,3,7,9,3,3,3,10,10,7,3,10,22,6,5,9,9,7,44,
%T A359421 3,5,3,6,10,3,15,5,3,6,6,15,15,12,20,3,11,3,3,10,7,14,18,30,5,9,21,3,
%U A359421 5,3,6,6,5,5,3,3,14,3,6,11,10,7,3,9,22,3,6,14
%N A359421 a(n) = number of abelian groups of order p^2 - 1, where p = prime(n).
%F A359421 a(n) = A000688(prime(n)^2-1) = A000688(A000040(n)^2-1) = A000688(A084920(n)).
%e A359421 For p = 5, p^2 - 1 = 24 = 2^3 * 3^1. The number of abelian groups of order 24 = (the number of partitions of 3)*(the number of partitions of 1) = 3*1 = 3.
%p A359421 a:= n-> mul(combinat[numbpart](i[2]), i=ifactors(ithprime(n)^2-1)[2]):
%p A359421 seq(a(n), n=1..79);  # _Alois P. Heinz_, Dec 31 2022
%t A359421 A000688[n_] := Times @@ (PartitionsP /@ FactorInteger[n][[All, 2]]);
%t A359421 a[n_] := A000688[Prime[n]^2 - 1];
%t A359421 Table[a[n], {n, 1, 79}] (* _Jean-François Alcover_, Feb 03 2025 *)
%o A359421 (Python)
%o A359421 from sympy import factorint, npartitions
%o A359421 from math import prod
%o A359421 def A359421(n): return prod(npartitions(d) for d in factorint(prime(n)**2-1).values()) # _Chai Wah Wu_, Jan 12 2023
%Y A359421 Subsequence of A000688.
%Y A359421 Cf. A000040, A000041, A084920.
%K A359421 nonn
%O A359421 1,2
%A A359421 _Ali Ramsey_, Dec 31 2022