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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 2 23:36 EST 2020. Contains 338898 sequences. (Running on oeis4.)