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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.
2
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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 16 2013
STATUS
approved