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

1,2

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, ..., .

Links

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

Example

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

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]

Prog

(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

Crossrefs

Cf. A005117, A006972, A056911, A253598.

Sequence in context: A285516 A319801 A339907 * A319181 A318718 A118749

Adjacent sequences: A255599 A255600 A255601 * A255603 A255604 A255605

Keyword

nonn

Author

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

Status

approved