A014090 Numbers that are not the sum of a square and a prime.

1, 10, 25, 34, 58, 64, 85, 91, 121, 130, 169, 196, 214, 226, 289, 324, 370, 400, 526, 529, 625, 676, 706, 730, 771, 784, 841, 1024, 1089, 1225, 1255, 1351, 1414, 1444, 1521, 1681, 1849, 1906, 1936, 2116, 2209, 2304, 2500, 2809, 2986, 3136, 3364, 3481, 3600

Offset: 1

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy
• nice

Comments

Sequence is infinite: if 2n-1 is composite then n^2 is in the sequence. (Proof: If n^2 = x^2 + p with p prime, then p = (n-x)(n+x), so n-x=1 and n+x=p. Hence 2n-1=p is prime, not composite.) - Dean Hickerson, Nov 27 2002

21679 is the last known nonsquare in this sequence. See A020495. - T. D. Noe, Aug 05 2006

A002471(a(n))=0; complement of A014089. - Reinhard Zumkeller, Sep 07 2008

There are no prime numbers in this sequence because at the very least they can be represented as p + 0^2. - Alonso del Arte, May 26 2012

Number of terms <10^k,k=0..8: 1, 8, 27, 75, 223, 719, 2361, 7759, ..., . - Robert G. Wilson v, May 26 2012

So far there are only 21 terms which are not squares and they are the terms of A020495. Those that are squares, their square roots are members of A104275. - Robert G. Wilson v, May 26 2012

Examples

			From _Alonso del Arte_, May 26 2012: (Start)
10 is in the sequence because none of 10 - p_i are square (8, 7, 5, 3) and none of 10 - b^2 are prime (10, 9, 6, 1); i goes from 1 to pi(10) or b goes from 0 to floor(sqrt(10)).
11 is not in the sequence because it can be represented as 3^2 + 2 or 0^2 + 11. (End)
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 115 terms from T. D. Noe)

Crossrefs

Cf. A020495, A104275.

Cf. A064233 (does not allow 0^2).

Programs

Mathematica

t={}; Do[k=0; While[k^2=n, AppendTo[t,n]], {n,25000}]; t (* T. D. Noe, Aug 05 2006 *)
max = 5000; Complement[Range[max], Flatten[Table[Prime[p] + b^2, {p, PrimePi[max]}, {b, 0, Ceiling[Sqrt[max]]}]]] (* Alonso del Arte, May 26 2012 *)
fQ[n_] := Block[{j = Sqrt[n], k}, If[ IntegerQ[j] && !PrimeQ[2j - 1], True, k = Floor[j]; While[k > -1 && !PrimeQ[n - k^2], k--]; If[k == -1, True, False]]]; Select[ Range[3600], fQ] (* Robert G. Wilson v, May 26 2012 *)