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%I #11 Dec 10 2016 20:15:47
%S 1,2,6,9,14,24,11,56,46,45,46,109,82,97,287,124,51,390,507,434,691,
%T 332,1105,898,676,359,1080,1259,659,1688,540,1146,4081,1672,3081,985,
%U 3975,2423,4460,6512,2779,10324,1820,5458,10273,8196,9177,7085,6462,5037
%N Let p and q be two prime numbers, not necessarily consecutive, such that q - p = 2n; then a(n) is the number of partitions of 2n into even numbers so that each partition corresponds to a consecutive prime difference pattern (k-tuple) and p <= A000230(n).
%C Partitions are counted with multiplicity and may overlap.
%e Only those partitions are counted that appear not later than prime A000230(n); n=7, d=14, A000230(7)=113; the number of solutions to p+14=q, with p and q both prime and p <= 113, is 11. These 11 (not necessarily distinct) partitions and their initial primes are as follows: 3[22424], 5[24242], 17[2462], 23[626], 29[2642], 47[662], 53[626], 59[2642], 83[68], 89[842], 113[14]=A000230(7).
%o (PARI) {for(n=1,50, c=0; p=2; done=0; until(done, if(isprime(p+2*n), c++; if(nextprime(p+1)-p==2*n, done=1; print1(c,","))); p=nextprime(p+1)))} \\ _Rick L. Shepherd_
%Y Cf. A000230, A079015-A079024.
%K nonn
%O 1,2
%A _Labos Elemer_, Jan 24 2003
%E Corrected and extended by _Rick L. Shepherd_, Sep 08 2003
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