A296304 Numbers whose absolute difference from a square is never a prime.

0, 169, 289, 625, 784, 1024, 1444, 1849, 2116, 2209, 3364, 3481, 3600, 3721, 3844, 4489, 5041, 5184, 5329, 5929, 6400, 7225, 7744, 8464, 8649, 8836, 10201, 10404, 10609, 10816, 11449, 11664, 11881, 12100, 13924, 14884, 15129, 15376, 16129, 16900, 17689, 18769

Offset

1,2

Comments

0 and the squares of numbers k such that 2k+1 and 2k-1 are not primes; i.e., 0 and the squares of the terms of A104278.

Links

Muniru A Asiru, Table of n, a(n) for n = 1..20000

Formula

a(1) = 0; for n > 1, A104278(n-1)^2.

Example

The absolute difference between any square j^2 and 169 is |j^2 - 169| = |(j-13)*(j+13)| = |j-13|*|j+13|, which cannot be a prime unless one of the two factors |j-13| and |j+13| is 1, i.e., j is -14, -12, 12, or 14; however, in each case, the other factor is nonprime (-27, -25, 25, or 27, respectively), so |j^2 - 169| is not a prime for any integer j. Thus, 169 is in the sequence.
49 - 6^2 = 49 - 36 = 13 (a prime), so 49 is not in the sequence.

Mathematica

Join[{0}, Select[Range[200], CompositeQ[2# + 1] && CompositeQ[2# - 1]&]^2] (* Jean-François Alcover, Dec 21 2017 *)

Prog

(GAP) o := [];; for n in [1..10^4] do if not IsPrime(2*n-1) and not IsPrime(2*n+1) then Add(o, n^2); fi; od;
sequence := Concatenation([0], o); # Muniru A Asiru, Jan 01 2018

Crossrefs

Cf. A104278.

Cf. A292990 (Numbers whose absolute difference from a triangular number is never a prime).

Sequence in context: A020249 A287391 A325421 * A156159 A099011 A330276

Adjacent sequences: A296301 A296302 A296303 * A296305 A296306 A296307

Keyword

nonn

Author

Jon E. Schoenfield, Dec 10 2017

Status

approved