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%I #17 Sep 08 2019 03:12:53
%S 1,1,0,1,2,0,1,1,4,8,0,8,42,28,140,616,836,180,1416,2542,10960,96048,
%T 242204,367587,923949,1145430,2622420,19081728,245846500,2934255428,
%U 6725485476,7722272142,26106311490,114470819132,331909473776,330258090272,4585951400436,37021666628450
%N Number of ways of pairing the squares of the numbers 1 to n with the squares of the numbers n+1 to 2n such that each pair sums to a prime. Because an odd square must always be added to an even square to obtain a prime, this sequence is the product of A077763 and A077764.
%C Apparently, for n>11, there seems always to be a pairing possible. Note that all primes have the 4k+1 form. By the 4k+1 theorem, such a prime has a unique representation as the sum of two squares.
%H Bert Dobbelaere, <a href="/A077762/b077762.txt">Table of n, a(n) for n = 1..50</a>
%H L. E. Greenfield and S. J. Greenfield, <a href="https://cs.uwaterloo.ca/journals/JIS/green.html">Some Problems of Combinatorial Number Theory Related to Bertrand's Postulate</a>, J. Integer Sequences, 1998, #98.1.2.
%F a(n) = permanent(m), where the n X n matrix m is defined by m(i,j) = 1 or 0, depending on whether i^2 + (j+n)^2 is prime or composite, respectively. - _T. D. Noe_, Feb 10 2007
%e a(5) = 2 because there are two ways: (1,4,9,16,25) + (36,49,100,81,64) = (37,53,109,97,89) and (1,4,9,16,25) + (100,49,64,81,36) = (101,53,73,97,61).
%t lst1*lst2 (* which are defined in A077763 and A077764 *)
%Y Cf. A000348, A070897, A077763, A077764.
%K nonn
%O 1,5
%A _T. D. Noe_, Nov 15 2002
%E More terms from _Bert Dobbelaere_, Sep 08 2019
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