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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
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
Sequence in context: A242498 A321927 A016194 * A261887 A037871 A037853
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 22 2015
STATUS
approved

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Last modified March 28 11:46 EDT 2024. Contains 371241 sequences. (Running on oeis4.)