

A258139


Number of ways to write n as p^2 + q with p and q both prime.


7



0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 2, 0, 1, 0, 2, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 0, 2, 0, 3, 1, 0, 0, 1, 0, 3, 1, 0, 1, 2, 0, 3, 0, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0, 3, 1, 0, 0, 2, 0, 2, 0, 0
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OFFSET

1,11


COMMENTS

Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5. Moreover, if n = 6*k + 2, then a(n) > 0 except for k = 0, 1, 12, 28, 102, 117, 168, 4079.
We have verified the conjecture for n up to 10^9.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


FORMULA

G.f.: (Sum_{k>=1} x^prime(k))*(Sum_{k>=1} x^(prime(k)^2)).  Ilya Gutkovskiy, Feb 05 2017


EXAMPLE

a(11) = 2 since 11 = 2^2 + 7 = 3^2 + 2 with 2, 3, 7 all prime.


MATHEMATICA

Do[r=0; Do[If[PrimeQ[nPrime[k]^2], r=r+1], {k, 1, PrimePi[Sqrt[n]]}]; Print[n, " ", r]; Continue, {n, 1, 100}]


CROSSREFS

Cf. A000040, A002375, A258140, A258141.
Sequence in context: A242498 A321927 A016194 * A261887 A037871 A037853
Adjacent sequences: A258136 A258137 A258138 * A258140 A258141 A258142


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 22 2015


STATUS

approved



