A166460 Numbers k such that k + (-1)^k is not prime.

0, 1, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94

Offset

1,3

Comments

This is the complement of A068499 (except that both include 1 as a term).

From Don Reble, Aug 31 2021: (Start)

Proof for all k except 0, 1, 3 with cases

(i) If k is odd and >=5, then k+1 = 2*x, 2 < x < k, k! = k*...*x*...*2*1

A068499: k+1 divides k! : absent

A166460: k-1 is even and composite : present

(ii) If k is even and k+1 is prime,

A068499: k+1 does not divide k! : present

A166460: k+1 is prime : absent

(iii) If k is even and k+1 = p^2 is the square of a (odd) prime, then k+1 >= 3p, k > 2p.

A068499: k! = k*...*2p*...*p*...*1;

k+1 divides k! : absent

A166460: k+1 is composite : present

(iv) If k is even and k+1 is composite but not the square of a prime, then there are two distinct factors x*y = k+1:

3 <= x < y = (k+1)/x < k.

A068499: k! = k*...*y*...*x*...*1:

k+1 divides k! : absent

A166460: k+1 is composite : present

(End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

0 + (-1)^0 = 1 is not prime, which adds 0 to the sequence.
5 + (-1)^5 = 4 is not prime, which adds 5 to the sequence.

Mathematica

Select[Range[0, 94], ! PrimeQ[# + (-1)^#] &] (* Michael De Vlieger, Sep 08 2021 *)

Crossrefs

Cf. A072668, A141468, A060462, A118742, A068499.

Sequence in context: A255922 A241028 A171097 * A118742 A122904 A104693

Adjacent sequences: A166457 A166458 A166459 * A166461 A166462 A166463

Keyword

nonn,easy

Author

Juri-Stepan Gerasimov, Oct 14 2009

Extensions

0 added by R. J. Mathar, Oct 21 2009

Status

approved