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

Author

Juri-Stepan Gerasimov, Oct 14 2009

Keywords

• nonn
• easy

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)

Examples

			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.
		

Links

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

Crossrefs

Cf. A002808, A072668, A141468, A060462, A118742, A068499.

Programs

Mathematica

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

Python

from sympy import composite
def A166460(n): return composite(n-1)-1 if n>2 else n-1 # Chai Wah Wu, Aug 27 2024

Formula

a(n) = A002808(n-1)-1 for n>2. - Chai Wah Wu, Aug 27 2024

Extensions

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