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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.
1
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
1,1
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.
EXAMPLE
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.
PROG
(PARI) isA328930(n) == (A328919(n)<A051903(n)) \\ See A328919 and A051903 for their program.
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Oct 31 2019
STATUS
approved