A255602 Numbers k which are odd and squarefree and have the property that k is either a prime number or for every prime p dividing k, p+1 is not divisible by any of the other prime factors of k.

1, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 35, 37, 39, 41, 43, 47, 53, 55, 57, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 93, 97, 101, 103, 107, 109, 111, 113, 115, 119, 127, 129, 131, 133, 137, 139, 143, 149, 151, 155

Offset: 1

Author

Tim Johannes Ohrtmann and Robert G. Wilson v, Feb 27 2015

Keywords

• nonn

Comments

A proper subset of A056911 and a proper subset of A005117. Any divisor of a Lucas-Carmichael number is in this sequence. It is not known whether every number in this sequence divides at least one Lucas-Carmichael number. All prime numbers except 2 are present. Composite numbers in the sequence include 21, 35, 39, 55, 57, 65, 77, 85, 93, 111, 115, 119, 129, 133, 143, 155, 161, 183, 185, 187, ..., .

Examples

			15 is not in the sequence since its two prime factors are 3 and 5, and 5+1 is divisible by 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A006972, A056911, A253598.

Programs

Mathematica

fQ[n_] := Block[{fi = FactorInteger@ n}, ffi = First@# & /@ fi; Times @@ (Last@# & /@ fi) == 1 && Min@ Flatten@ Table[ Mod[1 + ffi, i], {i, ffi}] > 0]; fQ[1] = True; fQ[2] = False; Select[ Range@ 190, fQ]

PARI

isok(n) = {if (! ((n % 2) && issquarefree(n)), return (0)); vpf = factor(n)[, 1]; for (i=1, #vpf, vpx = vpf[i]+1; for (j=1, #vpf, if (! (vpx % vpf[j]), return (0)); ); ); return (1); } \\ Michel Marcus, Mar 02 2015