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

1,1

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

Links

Table of n, a(n) for n=1..52.

Mathematica

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

Prog

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

Crossrefs

Cf. A002322, A002997, A306259.

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

Sequence in context: A023599 A087881 A063607 * A045956 A057339 A336552

Adjacent sequences: A306267 A306268 A306269 * A306271 A306272 A306273

Keyword

nonn

Author

Amiram Eldar and Thomas Ordowski, Feb 01 2019

Status

approved