A106564 Perfect squares which are not the difference of two primes.

25, 49, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1225, 1369, 1681, 1849, 2209, 2401, 2601, 2809, 3025, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 10201, 10609, 11449, 11881

Offset

1,1

Comments

Squares in A269345; see also the Mathematica code. - Waldemar Puszkarz, Feb 27 2016

It is conjectured (see A020483) that every even number is a difference of primes, and this is known to be true for even numbers < 10^11. If so,this sequence consists of the odd squares n such that n+2 is composite. - Robert Israel, Feb 28 2016

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

n^2 - A106546 with 0's removed.

Example

a(2)=49 because it is the second perfect square which is impossible to obtain subtracting a prime from another one.
64 is not in the sequence because 64=67-3 (difference of two primes).

Maple

remove(t -> isprime(t+2), [seq(i^2, i=1..1000, 2)]); # Robert Israel, Feb 28 2016

Mathematica

With[{lst=Union[(#[[2]]-#[[1]])&/@Subsets[Prime[Range[2000]], {2}]]}, Select[Range[140]^2, !MemberQ[lst, #]&]] (* Harvey P. Dale, Jan 04 2011 *)
Select[Range[1, 174, 2]^2, !PrimeQ[#+2]&]
Select[Select[Range[30000], OddQ[#]&& !PrimeQ[#]&& !PrimeQ[#+2]&], IntegerQ[Sqrt[#]]&] (* Waldemar Puszkarz, Feb 27 2016 *)

Prog

(PARI) for(n=1, 174, n%2==1&&!isprime(n^2+2)&&print1(n^2, ", ")) \\ Waldemar Puszkarz, Feb 27 2016
(Magma) [n^2: n in [1..150]| not IsPrime(n^2+2) and n mod 2 eq 1]; // Vincenzo Librandi, Feb 28 2016

Crossrefs

Cf. A020483, A106544-A106548, A106562-A106563, A106571, A106573-A106575, A106577.

Sequence in context: A374289 A109861 A348754 * A308177 A104777 A289829

Adjacent sequences: A106561 A106562 A106563 * A106565 A106566 A106567

Keyword

easy,nonn

Author

Alexandre Wajnberg, May 09 2005

Extensions

Extended by Ray Chandler, May 12 2005

Status

approved