A226599 Numbers which are the sum of two squared primes in exactly four ways (ignoring order).

10370, 10730, 11570, 12410, 13130, 19610, 22490, 25010, 31610, 38090, 38930, 39338, 39962, 40970, 41810, 55250, 55970, 59330, 59930, 69530, 70850, 73730, 76850, 77090, 89570, 98090, 98930, 103298, 118898, 125450, 126290, 130730, 135218, 139490

Offset: 1

Author

Henk Koppelaar, Jun 13 2013

Keywords

• nonn

Comments

It appears that all first differences are divisible by 24. - Zak Seidov, Jun 14 2013

Examples

			10370 = 13^2 + 101^2 = 31^2 + 97^2 = 59^2 + 83^2 = 71^2 + 73^2.
10730 = 11^2 + 103^2 = 23^2 + 101^2 = 53^2 + 89^2 = 67^2 + 79^2.
		

References

• Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A054735 (restricted to twin primes), A037073, A069496.

Cf. A045636 (sum of two squared primes is a superset).

Cf. A214511 (least number having n representations).

Cf. A225104 (numbers having at least three representations is a superset).

Cf. A226539, A226562 (sums decomposed in exactly two and three ways).

Programs

Maple

Prime2PairsSum := s -> select(x ->`if`(andmap(isprime, x), true, false),
   numtheory:-sum2sqr(s)):
for n from 2 to 10^6 do
  if nops(Prime2PairsSum(n)) = 4 then print(n, Prime2PairsSum(n)) fi;
od;

Mathematica

(* Assuming mod(a(n),24) = 2 *) Reap[ For[ k = 2, k <= 2 + 240000, k = k + 24, pr = Select[ PowersRepresentations[k, 2, 2], PrimeQ[#[[1]]] && PrimeQ[#[[2]]] &]; If[Length[pr] == 4 , Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 14 2013 *)

Formula

a(n) = p^2 + q^2; p, q are (not necessarily different) primes