A394390 - OEIS

%I #12 Mar 27 2026 22:53:15

%S 0,0,1,0,1,0,1,1,2,0,1,0,1,1,1,2,1,1,1,0,1,0,1,1,2,1,3,1,1,0,1,3,1,1,

%T 1,1,1,1,1,0,1,0,1,1,2,0,1,2,2,2,1,1,1,2,1,2,1,0,1,0,1,1,2,6,1,0,1,2,

%U 1,0,1,1,1,1,2,2,1,0,1,2,5,0,1,1,1,1,1,0,1,2,1,1,1

%N Number of nonisomorphic abelian groups of order n which do not appear as the group of units of the ring Z/kZ taken over every k such that phi(k) = n.

%C Every group of units is abelian.

%C If a(k) = 0, then k has a prime factorization k = s*2^r, where s is odd squarefree and r >= 1. See the Englezou link for a proof.

%C It also follows from the proof that A000688(k) = A380578(k) = A000041(r) for those k = s*2^r, since the prime signature of k is (r, 1, ..., 1); the equation follows by the partition product formula for the number of abelian groups of a given order. Thus a(k) = 0 only for group orders whose number of abelian groups is a partition number.

%H Miles Englezou, <a href="/A394390/b394390.txt">Table of n, a(n) for n = 1..10000</a>

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%H Miles Englezou, <a href="/A394390/a394390.txt">Proof</a>

%F a(n) = A000688(n) - A380578(n).

%e a(2) = 0 since C_2 is the only group of order 2 and C_2 appears as the group of units of the rings Z/3Z, Z/4Z, and Z/6Z, which are all the rings Z/kZ such that phi(k) = 2.

%e a(3) = 1 since C_3 is the only group of order 3 and C_3 does not appear as the group of units for any ring Z/kZ because phi(k) = 3 has no solution.

%e a(8) = 1 since of the 3 abelian groups of order 8, only C_8 does not appear as the group of units of the ring Z/kZ for any k such that phi(k) = 8.

%e a(16) = 2 since of the 5 abelian groups of order 16, only C_2 x C_2 x C_2 x C_2 and C_4 x C_4 do not appear as the group of units of the ring Z/kZ for any k such that phi(k) = 16.

%o (PARI) a(n) = my(f = factor(n)[,2]); prod(i = 1, #f, numbpart(f[i])) - #Set(apply(x -> znstar(x)[2], invphi(n))) \\ [using _Max Alekseyev_'s invphi.gp; see link]

%Y Cf. A000041, A000688, A380578.

%K nonn

%O 1,9

%A _Miles Englezou_, Mar 19 2026