A289337 Composite numbers (pseudoprimes) n, that are not Carmichael numbers, such that A000670(n-1) == 0 (mod n).

25, 125, 325, 451, 1561, 4089, 7107, 8625, 12025

Offset: 1

Author

Amiram Eldar, Jul 02 2017

Keywords

• nonn
• more

Comments

I. J. Good proved that A000670(k*(p-1)) == 0 (mod p) for all k >= 1 and prime p. Therefore the congruence A000670(n-1) == 0 (mod n) holds for all primes and Carmichael numbers. This sequence consist of the other composite numbers for which the congruence holds.

Examples

			A000670(24) = 2958279121074145472650648875 is divisible by 25 and 25 is not a prime, nor a Carmichael number.
		

Links

• I. J. Good, The number of orderings of n candidates when ties are permitted, Fibonacci Quarterly, Vol. 13 (1975), pp. 11-18.

Crossrefs

Cf. A000670, A002997, A289338.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]*a[n - k], {k, 1, n}]; carmichaelQ[n_]:=(Mod[n, CarmichaelLambda[n]] == 1); seqQ[n_] := !carmichaelQ[n] && Divisible[a[n-1],n]; Select[Range[2,500],seqQ]