# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a236531 Showing 1-1 of 1 %I A236531 #16 Apr 06 2014 10:49:32 %S A236531 0,0,1,2,2,2,2,3,2,3,1,4,2,3,4,1,3,2,3,5,2,4,3,2,4,1,5,4,3,5,3,3,4,3, %T A236531 7,5,4,7,1,7,1,5,8,3,8,5,5,5,3,9,6,6,7,4,6,3,5,8,6,7,5,6,4,5,7,7,6,5, %U A236531 4,4,6,5,7,6,9,3,5,5,5,6,5,8,5,5,6,5,7,4,5,10,3,7,5,6,3,4,7,5,6,6 %N A236531 a(n) = |{0 < k < n: {6*k -1 , 6*k + 1} and {prime(n-k), prime(n-k) + 2} are both twin prime pairs}|. %C A236531 Conjecture: (i) a(n) > 0 for all n > 2. %C A236531 (ii) If n > 3 is neither 11 nor 125, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1, prime(m) + 2 and 3*prime(m) - 10 are all prime. %C A236531 (iii) Any integer n > 458 can be written as p + q with q > 0 such that {p, p + 2} and {prime(q), prime(q) + 2} are both twin prime pairs. %C A236531 This is much stronger than the twin prime conjecture. We have verified part (i) of the conjecture for n up to 2*10^7. %H A236531 Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 %H A236531 Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014 %e A236531 a(11) = 1 since {6*1 - 1, 6*1 + 1} = {5, 7} and {prime(10), prime(10) + 2} = {29, 31} are both twin prime pairs. %e A236531 a(16) = 1 since {6*3 - 1, 6*3 + 1} = {17, 19} and {prime(13), prime(13) + 2} = {41, 43} are both twin prime pairs. %t A236531 p[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1] %t A236531 q[n_]:=PrimeQ[Prime[n]+2] %t A236531 a[n_]:=Sum[If[p[k]&&q[n-k],1,0],{k,1,n-1}] %t A236531 Table[a[n],{n,1,100}] %Y A236531 Cf. A000040, A001359, A002822, A006512, A199920. %K A236531 nonn %O A236531 1,4 %A A236531 _Zhi-Wei Sun_, Jan 27 2014 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE