A240712 Number of decompositions of 2n into an unordered sum of two terms of A240710.

0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 3, 4, 2, 2, 4, 2, 3, 5, 3, 3, 5, 2, 4, 6, 2, 4, 6, 2, 4, 6, 4, 4, 7, 4, 4, 8, 4, 4, 9, 3, 5, 7, 3, 5, 8, 4, 5, 8, 5, 6, 10, 5, 6, 12, 4, 5, 10, 3, 6, 9, 5, 5, 8, 6, 7, 11, 6, 5, 12, 3, 7, 11, 5, 7, 10, 5, 5, 13, 8

Offset: 1

Author

Lei Zhou, Apr 10 2014

Keywords

• nonn
• easy

Comments

a(n) differs from A171611 beginning at term a(264). To show the difference, the first 270 terms are listed.

Conjecture: a(n) > 0 for all n > 4.

This is a much stronger version of the Goldbach Conjecture.

Examples

			For n < 264, please refer to examples at A171611.
For n = 264, 2n=528. A240710 has terms {5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521} up to 528, where prime number 523 < 528 is not in the set, such that 528 = 5 + 523 is not counted in this sequence but is counted in A171611. So a(264) = A171611(264)-1 = 25-1 = 24.
		

Links

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

Crossrefs

Cf. A000040, A002375, A171611, A240710.

Programs

Mathematica

a240710 = {5}; Table[s = 2*n; While[a240710[[-1]] < s, p = a240710[[-1]]; While[p = NextPrime[p]; ok = 0; a1 = p - 12; a2 = p - 6; a3 = p + 6; a4 = p + 12; If[a1 > 0, If[PrimeQ[a1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2], ok = 1]]; If[PrimeQ[a3], ok = 1]; If[PrimeQ[a4], ok = 1]; ok == 0]; AppendTo[a240710, p]]; pos = 0; ct = 0; While[pos++; pos <= Length[a240710], p = a240710[[pos]]; If[p <= n, If[MemberQ[a240710, s - p], ct++]]]; ct, {n, 1, 270}]