|
%I #19 Mar 07 2021 00:56:09
%S 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,
%T 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,
%U 4,6,5,7,4,2,10,2,4,7,1,6,7,1,4,10,7,3,8
%N Number of decompositions of 2n into a sum of two primes p1 < p2 such that p2-p1 is between a pair of sexy primes.
%C "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.
%C The restriction is therefore p_a < p2 - p1 < p_a + 6 for p_a from A023201.
%C Conjecture: when n>=7, a(n)>0.
%C The products of sexy prime pairs are listed in A111192.
%H Lei Zhou, <a href="/A254690/b254690.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SexyPrimes.html">Sexy Primes</a>. [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].
%H Lei Zhou, <a href="/A254690/a254690.jpg">Plot of a(n)</a> for n <= 20000.
%e 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.
%e 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.
%e 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.
%t 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}]
%Y Cf. A023201, A045917, A111192.
%K nonn,easy
%O 1,8
%A _Lei Zhou_, Feb 05 2015
%E Edited by _Wolfdieter Lang_, Feb 20 2015
|
