OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 0. Similarly, for any integer n > 2, there is a permutation tau of {1,...,n} such that k*tau(k) - 1 is prime for every k = 1,...,n.
(ii) For any integer n > 2, there is a permutation tau of {1,...,n} such that k + tau(k) - 1 and k + tau(k) + 1 are twin prime for every k = 1,...,n.
Obviously, part (ii) of this conjecture implies the twin prime conjecture. P. Bradley proved in arXiv:1809.01012 that for any positive integer n there is a permutation tau of {1,...,n} such that k + tau(k) is prime for every k = 1,...,n.
LINKS
Paul Bradley, Prime number sums, arXiv:1809.01012 [math.GR], 2018.
Zhi-Wei Sun, Primes arising from permutations, Question 315259 on Mathoverflow, Nov. 14, 2018.
Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, arXiv:1811.10503 [math.CO], 2018.
EXAMPLE
a(3) = 1, and (1,3,2) is a permutation of {1,2,3} with 1*1 + 1 = 2, 2*3 + 1 = 7 and 3*2 + 1 = 7 all prime.
a(5) = 1, and (1,5,4,3,2) is a permutation of {1,2,3,4,5} with 1*1 + 1 = 2, 2*5 + 1 = 11, 3*4 + 1 = 13, 4*3 + 1 = 13 and 5*2 + 1 = 11 all prime.
MATHEMATICA
V[n_]:=V[n]=Permutations[Table[i, {i, 1, n}]]
tab={}; Do[r=0; Do[Do[If[PrimeQ[i*Part[V[n], k][[i]]+1]==False, Goto[aa]], {i, 1, n}]; r=r+1; Label[aa], {k, 1, n!}]; tab=Append[tab, r], {n, 1, 11}]
PROG
(PARI) a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i*j + 1))); \\ Jinyuan Wang, Jun 13 2020
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Zhi-Wei Sun, Nov 14 2018
EXTENSIONS
a(12)-a(26) from Alois P. Heinz, Nov 17 2018
a(27)-a(30) from Jinyuan Wang, Jun 13 2020
a(31)-a(36) from Vaclav Kotesovec, Aug 19 2021
STATUS
approved