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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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei 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[n-Prime[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: A153172 A242498 A016194 * A261887 A037871 A037853

Adjacent sequences:  A258136 A258137 A258138 * A258140 A258141 A258142

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 22 2015

STATUS

approved

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Last modified October 21 20:51 EDT 2018. Contains 316428 sequences. (Running on oeis4.)