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

%I A226599 #17 Sep 20 2018 00:30:43
%S A226599 10370,10730,11570,12410,13130,19610,22490,25010,31610,38090,38930,
%T A226599 39338,39962,40970,41810,55250,55970,59330,59930,69530,70850,73730,
%U A226599 76850,77090,89570,98090,98930,103298,118898,125450,126290,130730,135218,139490
%N A226599 Numbers which are the sum of two squared primes in exactly four ways (ignoring order).
%C A226599 It appears that all first differences are divisible by 24. - _Zak Seidov_, Jun 14 2013
%D A226599 Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.
%H A226599 T. D. Noe, <a href="/A226599/b226599.txt">Table of n, a(n) for n = 1..10000</a>
%F A226599 a(n) = p^2 + q^2; p, q are (not necessarily different) primes
%e A226599 10370 = 13^2 + 101^2 = 31^2 + 97^2 = 59^2 + 83^2 = 71^2 + 73^2.
%e A226599 10730 = 11^2 + 103^2 = 23^2 + 101^2 = 53^2 + 89^2 = 67^2 + 79^2.
%p A226599 Prime2PairsSum := s -> select(x ->`if`(andmap(isprime, x), true, false),
%p A226599    numtheory:-sum2sqr(s)):
%p A226599 for n from 2 to 10^6 do
%p A226599   if nops(Prime2PairsSum(n)) = 4 then print(n, Prime2PairsSum(n)) fi;
%p A226599 od;
%t A226599 (* 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 *)
%Y A226599 Cf. A054735 (restricted to twin primes), A037073, A069496.
%Y A226599 Cf. A045636 (sum of two squared primes is a superset).
%Y A226599 Cf. A214511 (least number having n representations).
%Y A226599 Cf. A225104 (numbers having at least three representations is a superset).
%Y A226599 Cf. A226539, A226562 (sums decomposed in exactly two and three ways).
%K A226599 nonn
%O A226599 1,1
%A A226599 _Henk Koppelaar_, Jun 13 2013