A254690 Number of decompositions of 2n into a sum of two primes p1 < p2 such that p2-p1 is between a pair of sexy primes.

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

Offset: 1

Author

Lei Zhou, Feb 05 2015

Keywords

• nonn
• easy

Comments

"A pair of sexy primes" is defined as two primes p_a < p_b such that p_b = p_a + 6, with p_a from A023201. See the Weisstein link.

The restriction is therefore p_a < p2 - p1 < p_a + 6 for p_a from A023201.

Conjecture: when n>=7, a(n)>0.

The products of sexy prime pairs are listed in A111192.

Examples

			n=7, 2n=14=3+11. 11-3=8, 5<8<11 where {5, 11} is a pair of sexy primes. So a(7)=1.
n=8, 2n=16=3+13=5+11. 13-3=10, 5<10<11; 11-5=6, 5<6<11, where {5, 11} is a pair of sexy primes: two cases found, so a(8)=2.
n=17, 2n=34=3+31=5+29=11+23. 31-3=28, 23<28<29; 29-5=24, 23<24<29; 23-11=12, 7<12<13; where {23,29} and {7,13} are sexy prime pairs: three cases found, so a(17)=3.
		

Links

• Lei Zhou, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].
• Lei Zhou, Plot of a(n) for n <= 20000.

Crossrefs

Cf. A023201, A045917, A111192.

Programs

Mathematica

Table[e = 2 n; ct = 0; p1 = 1; While[p1 = NextPrime[p1]; p1 < n, p2 = e - p1; If[PrimeQ[p2], c = p2 - p1; If[c >= 6, found = 0; Do[If[PrimeQ[c - i] && PrimeQ[c + 6 - i], found = 1], {i, 1, 5, 2}]; If[found == 1, ct++]]]]; ct, {n, 1, 100}]

Extensions

Edited by Wolfdieter Lang, Feb 20 2015