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A258141
Number of ways to write n as p^2 + q with p, q and 2*p + 3 all prime.
6
0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0
OFFSET
1,27
COMMENTS
Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5.
We have verified this for n up to 10^8. See also A258139 for a weaker version of this conjecture.
The conjecture is somewhat similar to Goldbach's Conjecture. It implies that there are infinitely many primes p with 2*p + 3 prime.
EXAMPLE
a(11) = 1 since 11 = 2^2 + 7 with 2, 7 and 2*2 + 3 all prime.
MATHEMATICA
Do[r=0; Do[If[PrimeQ[2Prime[k]+3]&&PrimeQ[n-Prime[k]^2], r=r+1], {k, 1, PrimePi[Sqrt[n]]}]; Print[n, " ", r]; Continue, {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 22 2015
STATUS
approved