A237710 Least prime p < n with pi(n-p) a square, or 0 if such a prime p does not exist.

0, 0, 2, 2, 3, 5, 5, 7, 2, 2, 2, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 2, 2, 2, 2, 2, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 2, 2, 2, 2, 2, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13

Offset

1,3

Comments

According to the conjecture in A237706, a(n) should be positive for all n > 2.

Links

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(5) = 3 since pi(5-3) = 1^2, but pi(5-2) = 2 is not a square.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
q[n_]:=SQ[PrimePi[n]]
Do[Do[If[q[n-Prime[k]], Print[n, " ", Prime[k]]; Goto[aa]], {k, 1, PrimePi[n-1]}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 100}]
lp[n_]:=Module[{p=2}, While[!IntegerQ[Sqrt[PrimePi[n-p]]], p=NextPrime[p]]; p]; Join[{0, 0}, Array[ lp, 80, 3]] (* Harvey P. Dale, Jan 28 2024 *)

Crossrefs

Cf. A000040, A000290, A000720, A237706.

Sequence in context: A316185 A131429 A105605 * A090473 A113636 A262463

Adjacent sequences: A237707 A237708 A237709 * A237711 A237712 A237713

Keyword

nonn

Author

Zhi-Wei Sun, Feb 12 2014

Status

approved