A324050 Numbers satisfying Korselt's criterion: squarefree numbers n such that for every prime divisor p of n, p-1 divides n-1.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359

Offset: 1

Author

Antti Karttunen, Feb 14 2019

Keywords

• nonn
• easy

Comments

Numbers n such that A002322(n) divides n-1.

Numbers that satisfy Korselt's criterion (see Mathworld and Wikipedia-links).

Union of {1}, primes and Carmichael numbers, A002997.

Differs from A008578 for the first time at term a(104) = 561, which (being a composite) is the first Carmichael number present.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Korselt's Criterion
• Wikipedia, Carmichael numbers, Korselt's criterion

Crossrefs

Union of A008578 and A002997. Intersection of A005117 and A087441. Subsequence of A015919.

Cf. A002322.

Programs

Mathematica

okQ[n_] := SquareFreeQ[n] && AllTrue[FactorInteger[n][[All, 1]], Divisible[n-1, #-1]&];
{1} ~Join~ Select[Range[2, 1000], okQ] (* Jean-François Alcover, Oct 22 2019 *)

PARI

isA324050(n) = if(1==n,1, my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); (1)); \\ After program Korselt in A002997
for(n=1,561,if(isA324050(n), print1(n,", ")));

PARI

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
isA324050(n) = (0==((n-1)%A002322(n)));