[seqfan] A166460, A068499
Don Reble
djr at nk.ca
Tue Aug 31 09:52:12 CEST 2021
Seqfans:
> %C A166460 Is this the complement of A068499? [R. J. Mathar]
Yes, except that they both have "1".
If N is odd and at least 5, then N+1 = 2*a, 2 < a < N,
N! = N*...*a*...*2*1
A068499: N+1 divides N! : absent
A166460: N-1 is even and composite : present
If N is even and N+1 is prime,
A068499: N+1 does not divide N! : present
A166460: N+1 is prime : absent
If N is even and N+1 = p^2 is the square of a (odd) prime,
then N+1 >= 3p, N > 2p.
A068499: N! = N*...*2p*...*p*...*1;
N+1 divides N! : absent
A166460: N+1 is composite : present
If N is even and N+1 is composite but not the square of a prime,
then there are two distinct factors a*b = N+1:
3 <= a < b = (N+1)/a < N.
A068499: N! = N*...*b*...*a*...*1:
N+1 divides N! : absent
A166460: N+1 is composite : present
Those cover all Ns except 0,1,3.
--
Don Reble djr at nk.ca
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