A240708 Number of decompositions of 2n into an unordered sum of two terms of A240699.

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, 7, 9, 6, 5, 8, 7, 8, 11, 6, 5, 12, 4, 8, 11, 5, 8, 10, 5, 6, 13, 9, 6, 11, 7, 7, 14, 6, 8, 13, 5, 8, 11, 7, 9, 13, 8, 9, 14, 7, 7, 19, 6, 7, 12, 6, 9

Offset

1,5

Comments

The first different term of this sequence to A002375 is a(107).

Conjecture: for n >= 3, this sequence is always positive.

This is a stronger version of the Goldbach Conjecture.

Links

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

Example

For n <= 106, refer to examples in A002375.
For n = 107, 2n=214. A240699 up to 214 gives {3, 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}.  We have 214 = 17+197 = 23+191 = 41+173 = 47+167 = 83+131 = 101+113 = 107+107. Seven instances found. So a(107)=7.
Where as for A002375, there is one more instance as 3+211, however 211 is not a term in A240699.

Mathematica

a240699 = {3}; Table[s = 2*n; While[a240699[[-1]] < s, p = a240699[[-1]]; While[p = NextPrime[p]; ((NextPrime[p] - p) > 6) && (6 < (p - NextPrime[p, -1]))]; AppendTo[a240699, p]]; pos = 0; ct = 0; While[pos++; pos <= Length[a240699], p = a240699[[pos]]; If[p <= n, If[MemberQ[a240699, s - p], ct++]]]; ct, {n, 1, 110}]

Crossrefs

Cf. A000040, A002375, A045917, A240699.

Sequence in context: A254610 A002375 A045917 * A235645 A325357 A294107

Adjacent sequences: A240705 A240706 A240707 * A240709 A240710 A240711

Keyword

nonn,easy

Author

Lei Zhou, Apr 10 2014

Status

approved