A328930 Numbers N such that A328919(N) < A051903(N); numbers N such that {sigma_k(N) mod N: k >= m} is purely periodic with some m < e, where e is the maximal exponent in prime factorization of N.

12, 18, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 68, 76, 80, 84, 88, 90, 92, 96, 99, 104, 108, 112, 116, 120, 124, 126, 132, 136, 140, 148, 152, 153, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 192, 198, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 240, 244, 248

Offset: 1

Author

Jianing Song, Oct 31 2019

Keywords

• nonn

Comments

It is easy to see that {sigma_k(N) mod N: k >= A051903(N)} is purely periodic.

All terms are nonsquarefree: if N is squarefree and N is here, then A328919(N) < A051903(N) = 1, so A328919(N) = 0. By the property mentioned in A328919, a necessary condition is that for every prime p dividing N, write N = p*s, we have p divides d(s), d = A000005. But d(s) is a power of 2, so N = 2, and 2 is not here.

Although it seems that for most N we have A328919(N) = A051903(N), this sequence is infinite. See A328934 for more information.

Examples

			A328919(12) = 0, while A051903(12) = 2, so 12 is a term.
A328919(24) = 1, while A051903(24) = 3, so 24 is a term.
If N = p^e for prime p, then A328919(p^e) = A051903(p^e) = e. So this sequence and A000961 have empty intersection.
		

Crossrefs

Cf. A328919, A051903, A328934, A000961, A000005.

Programs

PARI

isA328930(n) == (A328919(n)<A051903(n)) \\ See A328919 and A051903 for their program.