A316111 a(n) is the smallest k > 1 such that gcd(k, n^k - n) = 1, for n > 1.

35, 35, 77, 77, 143, 55, 55, 77, 119, 119, 35, 55, 187, 143, 77, 35, 35, 77, 143, 247, 95, 35, 77, 77, 77, 55, 55, 143, 77, 77, 35, 35, 247, 143, 143, 35, 35, 77, 77, 143, 55, 95, 119, 119, 77, 35, 55, 143, 143, 77, 35, 35, 119, 299, 221, 55, 35, 77, 77, 77, 55, 55, 187, 119

Offset: 2

Author

Thomas Ordowski, Aug 13 2018

Keywords

• nonn

Comments

Conjecture: all the terms are in A121707. If k is a term, then k is an "anti-Carmichael number": p-1 does not divide k-1 for every prime p dividing k.

The sequence is unbounded, since a(m!) > m.

Prediction: a(n) < n for all sufficiently large n.

GCD(n, a(n)) = 1. a(n) is odd. Is a(n) squarefree? - David A. Corneth, Aug 13 2018

Crossrefs

Cf. A121707, A267999.

Programs

PARI

a(n) = {my(k=2); while (gcd(k, n^k - n) != 1, k++); k;} \\ Michel Marcus, Aug 13 2018

PARI

a(n) = {my(k=3); while (gcd(k, n^k - n) != 1, k+=2; while(gcd(n, k) > 1, k+=2)); k} \\ David A. Corneth, Aug 13 2018

Extensions

More terms from Michel Marcus, Aug 13 2018