A319484 a(n) is the smallest k > 1 such that n^k == n (mod k) and gcd(k, b^k-b) = 1 for some b <> n.

35, 35, 7957, 16531, 1247, 4495, 35, 817, 2501, 697, 55, 55, 143, 221, 35, 35, 1247, 493, 221, 95, 35, 35, 77, 253, 115, 403, 247, 247, 203, 35, 155, 155, 697, 187, 35, 35, 35, 589, 221, 95, 533, 35, 287, 77, 55, 55, 115, 221, 329, 35, 35, 221, 221, 689, 55, 35, 35

Offset: 0

Author

Thomas Ordowski, Oct 26 2018

Keywords

• nonn

Comments

a(n) is the smallest k > 1 such that n^k == n (mod k) and p-1 does not divide k-1 for every prime p dividing k, see A121707.

We have A000790(n) < a(n) <= A316940(n) for n > 0.

It seems that the sequence is unbounded like A316940.

The term a(5) = 4495 = 5*29*31 is not semiprime.

Examples

			a(6) = 35 since 6^35 == 6 (mod 35) and 35 = 5*7 is the smallest "anti-Carmichael number": 5-1 does not divide 7-1. We have gcd(35,2^35-2) = 1.
		

Crossrefs

Cf. A000790, A121707, A316940.

Programs

PARI

isac(n) = {my(f = factor(n)[,1]); for (i=1, #f, if (((n-1) % (f[i]-1)) == 0, return (0));); return (1);}
isok(n,k) = {if (Mod(n, k)^k != Mod(n, k), return (0)); return (isac(k));}
a(n) = {my(k=2); while (!isok(n,k), k++); return (k);} \\ Michel Marcus, Oct 27 2018

Extensions

More terms from Michel Marcus, Oct 26 2018