A230362 Least prime p with 2*p^2 - 1 and 2*(n-p)^2 -1 both prime, or 0 if such a prime p does not exist.

3, 13, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 7, 2, 2, 3, 2, 2, 3, 7, 2, 3, 7, 11, 13, 7, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 3, 2, 3, 7, 7, 2, 3, 11, 2, 3, 7, 2, 2, 2, 3, 43, 7, 7

Offset

1,1

Comments

Conjecture: 0 < a(n) < sqrt(2n)*(log n) except for n = 1, 2, 3, 232, 1478, 6457.

By the conjecture in the comments in A230351, 0 < a(n) < n for all n > 3.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(12) = 2 since 2*2^2 - 1 and 2*(12-2)^2 - 1 = 199 are both prime.

Mathematica

Do[Do[If[PrimeQ[2Prime[i]^2-1]&&PrimeQ[2(n-Prime[i])^2-1], Print[n, " ", Prime[i]]; Goto[aa]], {i, 1, Max[13, PrimePi[n-1]]}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 70}]

Crossrefs

Cf. A000040, A066049, A106483, A230351.

Sequence in context: A073580 A340349 A326609 * A173203 A140445 A320039

Adjacent sequences: A230359 A230360 A230361 * A230363 A230364 A230365

Keyword

nonn

Author

Zhi-Wei Sun, Oct 16 2013

Status

approved