A230546 Least positive integer k <= n such that 2*k^2-1 is a prime and n - k is a square, or 0 if such an integer k does not exist.

0, 2, 2, 3, 4, 2, 3, 4, 8, 6, 2, 3, 4, 10, 6, 7, 8, 2, 3, 4, 17, 6, 7, 8, 21, 10, 2, 3, 4, 21, 6, 7, 8, 18, 10, 11, 21, 2, 3, 4, 25, 6, 7, 8, 36, 10, 11, 39, 13, 25, 2, 3, 4, 18, 6, 7, 8, 22, 10, 11

Offset

1,2

Comments

By the conjecture in A230494, we should have a(n) > 0 for all n > 1.

Links

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

Example

a(4) = 3 since neither 4 - 1 = 3 nor 4 - 2 = 2 is a square, but 4 - 3 = 1 is a square and 2*3^2 - 1 = 17 is a prime.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[Do[If[PrimeQ[2k^2-1]&&SQ[n-k], Print[n, " ", k]; Goto[aa]], {k, 1, n}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 60}]
lpik[n_]:=Module[{k=1}, While[!PrimeQ[2k^2-1]||!IntegerQ[Sqrt[n-k]], k++]; k]; Join[{0}, Array[lpik, 60, 2]] (* Harvey P. Dale, Aug 04 2021 *)

Crossrefs

Cf. A000040, A000290, A066049, A230351, A230362, A230493, A230494.

Sequence in context: A236241 A127731 A159978 * A286617 A328446 A371306

Adjacent sequences: A230543 A230544 A230545 * A230547 A230548 A230549

Keyword

nonn

Author

Zhi-Wei Sun, Oct 23 2013

Status

approved