

A220431


Number of ways to write n=x+y (x>0, y>0) with 3x1, 3x+1 and xy1 all prime.


5



0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 6, 1, 3, 6, 4, 3, 3, 2, 3, 4, 3, 4, 2, 3, 3, 5, 4, 4, 7, 1, 2, 5, 1, 5, 7, 4, 2, 3, 7, 4, 7, 2, 4, 7, 4, 4, 5, 2, 5, 8, 4, 3, 3, 5, 2, 8, 5, 4, 3, 10, 7, 8, 2, 3, 5, 5, 3, 6, 3, 3, 14, 4, 3, 12, 3, 7, 7, 5, 6, 8, 7, 5, 9, 9, 4, 4, 3, 6, 10, 8
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OFFSET

1,5


COMMENTS

Conjecture: a(n)>0 for all n>3.
This has been verified for n up to 10^8, and it is stronger than A. Murthy's conjecture related to A109909.
The conjecture implies the twin prime conjecture for the following reason: If x_1<...<x_k are positive integers and q_1,...,q_k are distinct primes greater than x_k, then by the Chinese Remainder Theorem there are infinitely many positive integers n such that x_i(nx_i) == 1 (mod q_i).
ZhiWei Sun also made some similar conjectures. For example, any integer n>2 not equal to 63 can be written as x+y (x>0, y>0) with 2x1, 2x+1 and 2xy+1 all prime.


LINKS



EXAMPLE

a(22)=1 since 22=4+18 with 3*41, 3*4+1 and 4*181 all prime.


MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[3k1]==True&&PrimeQ[3k+1]==True&&PrimeQ[k(nk)1]==True, 1, 0], {k, 1, n1}]
Do[Print[n, " ", a[n]], {n, 1, 1000}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



