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A000348 Number of ways to pair up {1^2, 2^2, ..., (2n)^2 } so sum of each pair is prime. 1
1, 1, 2, 4, 12, 9, 72, 160, 428, 2434, 3011, 10337, 126962, 264182, 783550, 5004266, 34340141, 176302123, 1188146567, 4457147441, 7845512385, 132253267889, 1004345333251, 3865703506342, 40719018858150, 213982561376958, 1266218151414286, 10976172953868304, 59767467676582641, 512279001476451101, 6189067229056357433 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..31.

B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

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 (2i)^2+(2j-1)^2 is prime or composite, respectively. - T. D. Noe, Feb 10 2007

MATHEMATICA

a[n_] := Permanent[Table[Boole[PrimeQ[(2*i)^2 + (2*j - 1)^2]], {i, 1, n}, {j, 1, n}]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 22}] (* Jean-Fran├žois Alcover, Jan 06 2016, after T. D. Noe *)

PROG

(PARI) permRWNb(a)=n=matsize(a)[1]; if(n==1, return(a[1, 1])); sg=1; nc=0; in=vectorv(n); x=in; x=a[, n]-sum(j=1, n, a[, j])/2; p=prod(i=1, n, x[i]); for(k=1, 2^(n-1)-1, sg=-sg; j=valuation(k, 2)+1; z=1-2*in[j]; in[j]+=z; nc+=z; x+=z*a[, j]; p+=prod(i=1, n, x[i], sg)); return(2*(2*(n%2)-1)*p)

for(n=1, 24, a=matrix(n, n, i, j, isprime((2*i)^2+(2*j-1)^2)); print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

CROSSREFS

Cf. A000341.

Sequence in context: A278225 A186118 A181817 * A168342 A141668 A087796

Adjacent sequences:  A000345 A000346 A000347 * A000349 A000350 A000351

KEYWORD

nonn,nice

AUTHOR

S. J. Greenfield (greenfie(AT)math.rutgers.edu)

EXTENSIONS

a(11)-a(16) from David W. Wilson

a(17)-a(22) from T. D. Noe, Feb 10 2007

a(23)-a(24) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

More terms from Sean A. Irvine, Nov 14 2010

STATUS

approved

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Last modified March 23 21:31 EDT 2017. Contains 283985 sequences.