A306259 Composite numbers k such that 2^(k(k-1)) == 1 (mod k^2).

21, 105, 165, 205, 231, 273, 301, 341, 385, 465, 561, 609, 645, 651, 861, 889, 903, 1045, 1065, 1105, 1265, 1281, 1365, 1387, 1491, 1705, 1729, 1771, 1785, 1905, 2041, 2047, 2145, 2211, 2265, 2329, 2359, 2373, 2465, 2485, 2665, 2667, 2701, 2821, 3045, 3081, 3165, 3171, 3201, 3277

Offset

1,1

Comments

Composites k such that A002326((k^2-1)/2) divides k(k-1).

It contains all Fermat pseudoprimes to base 2, A001567.

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

filter:= k -> not isprime(k) and 2 &^ (k*(k-1)) mod (k^2) = 1:
select(filter, [$4..10000]); # Robert Israel, Feb 07 2019

Mathematica

Select[Range[3300], And[CompositeQ@ #, PowerMod[2, # (# - 1), #^2] == 1] &] (* Michael De Vlieger, Feb 03 2019 *)

Prog

(PARI) isok(k) = !isprime(k) && ((2^(k*(k-1)) % k^2) == 1); \\ Michel Marcus, Feb 01 2019

Crossrefs

Cf. A001567, A002326, A306270.

Sequence in context: A219297 A160082 A201468 * A069499 A068142 A126229

Adjacent sequences: A306256 A306257 A306258 * A306260 A306261 A306262

Keyword

nonn

Author

Thomas Ordowski, Feb 01 2019

Status

approved