|
|
A000341
|
|
Number of ways to pair up {1..2n} so sum of each pair is prime.
|
|
11
|
|
|
1, 2, 3, 6, 26, 96, 210, 1106, 3759, 12577, 74072, 423884, 2333828, 16736611, 99838851, 630091746, 4525325020, 38848875650, 342245714017, 3335164762941, 31315463942337, 241353231085002, 2350106537365732, 17903852593938447, 158065352670318614, 1815064841856534244, 20577063085601738871, 276081763499377227299
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
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+2j-1 is prime or composite, respectively. - T. D. Noe, Feb 10 2007
|
|
EXAMPLE
|
For n=4, there are 6 ways to pair up {1, 2, 3, 4, 5, 6, 7, 8} so that each pair sums to a prime:
1+2, 3+4, 5+8, 6+7
1+2, 3+8, 4+7, 5+6
1+4, 2+3, 5+8, 6+7
1+4, 2+5, 3+8, 6+7
1+6, 2+3, 4+7, 5+8
1+6, 2+5, 3+8, 4+7
|
|
MAPLE
|
f:= proc(n) local M;
M:= Matrix(n, n, (i, j) -> `if`(isprime(2*i+2*j-1), 1, 0));
LinearAlgebra:-Permanent(M)
end proc:
|
|
MATHEMATICA
|
a[n_] := Permanent[ Array[ Boole[ PrimeQ[2*#1 + 2*#2 - 1]] & , {n, n}]]; Table[an = a[n]; Print[an]; an, {n, 1, 20}] (* Jean-François Alcover, Oct 21 2011, after T. D. Noe, updated Feb 07 2016 *)
|
|
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+j)-1)); print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007
(PARI) a(n)=matpermanent(matrix(n, n, i, j, isprime(2*(i+j)-1))); \\ Martin Fuller, Sep 22 2023
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice
|
|
AUTHOR
|
S. J. Greenfield (greenfie(AT)math.rutgers.edu)
|
|
EXTENSIONS
|
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007
|
|
STATUS
|
approved
|
|
|
|