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A077762
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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.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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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.
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LINKS
| 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.
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FORMULA
| a(n)=permanent(m), where the n-by-n matrix m is defined m(i,j) = 1 or 0, depending on whether i^2+(j+n)^2 is prime or composite, respectively. - T. D. Noe (noe(AT)sspectra.com), Feb 10 2007
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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).
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MATHEMATICA
| lst1*lst2 (* which are defined in A077763 and A077764 *)
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CROSSREFS
| Cf. A000348, A070897, A077763, A077764.
Sequence in context: A144740 A049501 A102564 * A085496 A180160 A101661
Adjacent sequences: A077759 A077760 A077761 * A077763 A077764 A077765
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KEYWORD
| hard,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Nov 15 2002
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