|
|
A321610
|
|
Number of permutations tau of {1,...,n} such that k^2 + tau(k)^2 is prime for every k = 1,...,n.
|
|
7
|
|
|
1, 1, 1, 1, 1, 4, 0, 16, 4, 144, 64, 81, 256, 5184, 1600, 25600, 8100, 183184, 108900, 5924356, 342225, 9066121, 11356900, 106853569, 105698961, 16119349444, 1419933124, 69792129124, 14251584400, 613950602500, 304388337796, 25042678198756, 10080904401936, 1179245283899881, 1045903153861476, 31082438574307129
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,6
|
|
COMMENTS
|
Conjecture 1: The number a(n) is always a square, and a(n) = 0 only for n = 7.
Conjecture 2: For any positive integer n, there is a permutation tau of {1,...,n} such that k^2 + k*tau(k) + tau(k)^2 is prime for every k = 1,...,n.
|
|
LINKS
|
|
|
EXAMPLE
|
a(3) = 1, and (1,3,2) is a permutation of {1,2,3} with 1^2 + 1^2 = 2, 2^2 + 3^2 = 13 and 3^2 + 2^2 all prime.
a(5) = 1, and (1,3,2,5,4) is a permutation of {1,2,3,4,5} with 1^2 + 1^2 = 2, 2^2 + 3^2 = 13, 3^2 + 2^2 = 13, 4^2 + 5^2 = 41 and 5^2 + 4^2 = 41 all prime.
|
|
MATHEMATICA
|
V[n_]:=V[n]=Permutations[Table[i, {i, 1, n}]]
Do[r=0; Do[Do[If[PrimeQ[i^2+Part[V[n], k][[i]]^2]==False, Goto[aa]], {i, 1, n}]; r=r+1; Label[aa], {k, 1, n!}]; Print[n, " ", r], {n, 1, 11}]
|
|
PROG
|
(PARI) a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i^2 + j^2))); \\ Jinyuan Wang, Jun 13 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|