A077762 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.

1, 1, 0, 1, 2, 0, 1, 1, 4, 8, 0, 8, 42, 28, 140, 616, 836, 180, 1416, 2542, 10960, 96048, 242204, 367587, 923949, 1145430, 2622420, 19081728, 245846500, 2934255428, 6725485476, 7722272142, 26106311490, 114470819132, 331909473776, 330258090272, 4585951400436, 37021666628450

Offset

1,5

Comments

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.

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..50

L. E. Greenfield and S. J. Greenfield, Some Problems of Combinatorial Number Theory Related to Bertrand's Postulate, J. Integer Sequences, 1998, #98.1.2.

Formula

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

Example

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).

Mathematica

lst1*lst2 (* which are defined in A077763 and A077764 *)

Crossrefs

Cf. A000348, A070897, A077763, A077764.

Sequence in context: A247504 A306800 A235955 * A244677 A243986 A322838

Adjacent sequences: A077759 A077760 A077761 * A077763 A077764 A077765

Keyword

nonn

Author

T. D. Noe, Nov 15 2002

Extensions

More terms from Bert Dobbelaere, Sep 08 2019

Status

approved