%I #33 Dec 18 2016 22:45:15

%S 1,1,2,4,12,9,72,160,428,2434,3011,10337,126962,264182,783550,5004266,

%T 34340141,176302123,1188146567,4457147441,7845512385,132253267889,

%U 1004345333251,3865703506342,40719018858150,213982561376958,1266218151414286,10976172953868304,59767467676582641,512279001476451101,6189067229056357433

%N Number of ways to pair up {1^2, 2^2, ..., (2n)^2 } so sum of each pair is prime.

%H B. K. Agarwala and F. C. Auluck, <a href="/A000093/a000093.pdf">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

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

%t 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-François Alcover_, Jan 06 2016, after _T. D. Noe_ *)

%o (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)

%o 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

%Y Cf. A000341.

%K nonn,nice

%O 1,3

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

%E a(11)-a(16) from _David W. Wilson_

%E a(17)-a(22) from _T. D. Noe_, Feb 10 2007

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

%E More terms from _Sean A. Irvine_, Nov 14 2010