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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.

%I A230443 #20 Jun 02 2025 08:43:24
%S A230443 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,
%T A230443 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,
%U A230443 8,7,7,10,6,5,12,3,7,10,5,5,10,1,6,10,7,4
%N 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.
%C A230443 1. This is a tightly intensified version of Goldbach conjecture.
%C A230443 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.
%C A230443 2. The definition of "the number of runs in binary expansion of k" is from A005811.
%C A230443 3. The first difference of this sequence to A002375 is on a(26).
%H A230443 Lei Zhou, <a href="/A230443/b230443.txt">Table of n, a(n) for n = 2..10000</a>
%e A230443 n=2, 2n=4, 4=2+2, 2-2=0, A005811(0)=1 < 4, only one decomposition, so a(2)=1;
%e A230443 ...
%e A230443 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;
%e A230443 ...
%e A230443 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.
%t A230443 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
%t A230443 [rank<=4,ct++]]];ct,{seed,2,100}]
%Y A230443 Cf. A002372, A005811
%K A230443 nonn,base,easy
%O A230443 2,4
%A A230443 _Lei Zhou_, Oct 18 2013