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A306270
Composite numbers k such that b^(k(k-1)) == 1 (mod k^2) for every b coprime to k.
1
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
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
A190275 is a subsequence. - Thomas Ordowski, Jul 19 2020
Sequence in context: A023599 A087881 A063607 * A045956 A057339 A336552
KEYWORD
nonn
AUTHOR
Amiram Eldar and Thomas Ordowski, Feb 01 2019
STATUS
approved