A306270 Composite numbers k such that b^(k(k-1)) == 1 (mod k^2) for every b coprime to k.

4, 6, 12, 20, 21, 28, 42, 52, 60, 66, 84, 105, 156, 165, 186, 220, 231, 273, 276, 301, 364, 385, 420, 465, 506, 532, 561, 609, 645, 651, 660, 780, 804, 903, 946, 1036, 1045, 1065, 1092, 1105, 1204, 1265, 1281, 1365, 1491, 1540, 1705, 1716, 1729, 1771, 1806, 1860

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Feb 01 2019

Keywords

• nonn

Comments

These are composites k such that lambda(k^2) divides k(k-1), where lambda is the Carmichael function A002322.

Since lambda(p^2) = phi(p^2) = p(p-1), where p is a prime, then by Euler's theorem b^(p(p-1)) == 1 (mod p^2) for every b indivisible by p.

This sequence includes all Carmichael numbers A002997.

Conjecture: all semiprimes > 4 in this sequence are in A190275. - Thomas Ordowski, Jul 19 2020

The conjecture was verified up to 1063290841. - Amiram Eldar, Jul 19 2020

Crossrefs

Cf. A002322, A002997, A306259.

A190275 is a subsequence. - Thomas Ordowski, Jul 19 2020

Programs

Mathematica

Select[Range[2000], CompositeQ[#] && Divisible[#(#-1), CarmichaelLambda[#^2]] &]

PARI

isok(k) = (k!=1) && !isprime(k) && !(k*(k-1) % lcm(znstar(k^2)[2])); \\ Michel Marcus, Mar 12 2019