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

2,4

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).

Links

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

Example

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.

Mathematica

Table[ev=2*seed; ct=0; cp1=seed-1; While[cp1=NextPrime[cp1]; cp1<ev, cp2=ev-cp1; If[PrimeQ[cp2], test=cp1-cp2; rank=Length[Length/@Split[IntegerDigits[test, 2]]]; If
[rank<=4, ct++]]]; ct, {seed, 2, 100}]

Crossrefs

Cf. A002372, A005811

Sequence in context: A094570 A225638 A332656 * A254610 A002375 A045917

Adjacent sequences: A230440 A230441 A230442 * A230444 A230445 A230446

Keyword

nonn,base,easy

Author

Lei Zhou, Oct 18 2013

Status

approved