A230443 Number of decompositions of 2n into a sum of two primes p2 >= p1 such that the number of runs in binary expansion of p2-p1 is less than or equal to 4.

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

Offset: 2

Author

Lei Zhou, Oct 18 2013

Keywords

• nonn
• base
• easy

Comments

1. This is a tightly intensified version of Goldbach conjecture.

It is hypothesized that except for n=1402 and 27242, all other terms for n > 1 are greater than zero. Sequence tested up to 1 million without other zero elements.

2. The definition of "the number of runs in binary expansion of k" is from A005811.

3. The first difference of this sequence to A002375 is on a(26).

Examples

			n=2, 2n=4, 4=2+2, 2-2=0, A005811(0)=1 < 4, only one decomposition, so a(2)=1;
...
n=5, 2n=10, 10=5+5=3+7, 5-5=0, A005811(0)=1<4, 7-3=4, A005811(4)=2<4, so a(5)=2;
...
n=26, 2n=52, 52=5+47=11+41=23+29.  47-5=42, A005811(42)=6>4 [X]; 41-11=30, A005811(30)=2<4 [v]; 29-23=6, A005811(6)=2<4 [v]; so a(26)=2.
		

Links

• Lei Zhou, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A002372, A005811

Programs

Mathematica

Table[ev=2*seed;ct=0;cp1=seed-1;While[cp1=NextPrime[cp1];cp1