A258168 Number of ways to write n as floor((p^2+q)/5) with p and q both prime.

3, 4, 3, 4, 5, 4, 4, 4, 4, 7, 4, 5, 6, 4, 5, 5, 4, 5, 4, 3, 5, 6, 4, 6, 5, 6, 5, 5, 3, 6, 6, 7, 3, 7, 5, 8, 8, 5, 5, 9, 5, 4, 6, 7, 4, 7, 5, 6, 7, 5, 4, 5, 4, 7, 8, 6, 6, 8, 4, 8, 7, 5, 8, 7, 4, 7, 5, 7, 4, 6, 6, 13, 7, 7, 6, 8, 4, 10, 10, 9

Offset

1,1

Comments

Conjecture: Let n be any positive integer. Then a(n) > 0. Moreover, one of the four consecutive numbers 5*n, 5*n+1, 5*n+2, 5*n+3 can be written as p^2+q with p and q both prime.

It seems that there are infinitely many positive integers n such that none of n, n+1, n+2, n+3, n+4 has the form p^2 + q with p and q both prime.

See also A258141 for a similar conjecture.

Note that neither 3763 nor 5443 can be written as floor((p^2+q)/4) with p and q both prime.

Links

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

Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.

Example

a(1) = 3 since 1 = floor((2^2+2)/5) = floor((2^2+3)/5) = floor(2^2+5)/5) with 2, 3, 5 all prime.
a(2) = 4 since 2 = floor((2^2+7)/5) = floor((3^2+2)/5) = floor((3^2+3)/5) = floor((3^2+5)/5) with 2, 3, 5, 7 all prime.

Mathematica

Do[m=0; Do[If[PrimeQ[5n+r-Prime[k]^2], m=m+1], {r, 0, 4}, {k, 1, PrimePi[Sqrt[5n+r]]}]; Print[n, " ", m]; Continue, {n, 1, 80}]

Crossrefs

Cf. A000040, A258139, A258140, A258141.

Sequence in context: A100867 A220196 A128200 * A061988 A094151 A135800

Adjacent sequences: A258165 A258166 A258167 * A258169 A258170 A258171

Keyword

nonn

Author

Zhi-Wei Sun, May 22 2015

Status

approved