

A240713


Number of decompositions of 2n=p1+p2 (prime p1 <= p2), where at least one other such pair 2n=p3+p4 (prime p3 <= p4) exists such that p1p3= 6 or 12.


2



0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 3, 4, 2, 2, 4, 2, 3, 5, 3, 3, 5, 2, 4, 6, 2, 4, 5, 0, 4, 5, 4, 4, 7, 3, 4, 6, 3, 4, 9, 3, 5, 7, 3, 5, 7, 3, 5, 7, 5, 6, 10, 5, 6, 12, 2, 5, 10, 2, 6, 7, 4, 4, 4, 5, 7, 9, 6, 5, 11, 0, 6, 10, 3, 7, 8, 4, 4, 13, 8
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OFFSET

1,11


COMMENTS

It is conjectured that a(n)=0 only when n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 34, 76, 229.
The 2n decompositions counted in this sequence are a subset of 2n decompositions as of in Goldbach conjecture (A002375).
Per definition, all nonzero terms are greater than 1.


LINKS

Lei Zhou, Table of n, a(n) for n = 1..10000


EXAMPLE

For n = 11, 2n=22. 22 = 3 + 19 = 5 + 17 = 11 + 11. 511=6 so pair 5+17 and 11+11 are counted. So a(11)=2.
...
For n = 17, 2n=34. 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. 511=6, so 5+29 and 11+23 are counted. Also since 1117=6, 17+17 is also counted (where 11+23 is already counted). In case 517=12, both instances are already counted. So overall three instances are found. a(17)=3.


MATHEMATICA

Table[s = 2*n; ct = 0; p = 1; While[p = NextPrime[p]; p <= n, If[PrimeQ[s  p], ok = 0; a1 = p  12; b1 = s  a1; a2 = p  6; b2 = s  a2; a3 = p + 6; b3 = s  a3; a4 = p + 12; b4 = s  a4; If[a1 > 0, If[PrimeQ[a1] && PrimeQ[b1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2] && PrimeQ[b2], ok = 1]]; If[a3 <= n, If[PrimeQ[a3] && PrimeQ[b3], ok = 1]]; If[a4 <= n, If[PrimeQ[a4] && PrimeQ[b4], ok = 1]]; If[ok == 1, ct++]]]; ct, {n, 1, 85}]


CROSSREFS

Cf. A000040, A002375, A240712.
Sequence in context: A059963 A137934 A133738 * A111409 A125088 A226456
Adjacent sequences: A240710 A240711 A240712 * A240714 A240715 A240716


KEYWORD

nonn,easy


AUTHOR

Lei Zhou, Apr 10 2014


STATUS

approved



