A213202 - OEIS

%I #16 Apr 18 2013 02:45:13

%S 0,0,1,2,2,3,3,2,2,3,1,2,2,2,3,2,1,3,2,1,3,1,3,5,2,2,3,2,3,4,4,4,2,3,

%T 3,3,3,2,1,2,4,5,4,4,4,2,3,3,4,4,3,2,1,4,6,6,4,4,4,4,4,4,4,2,3,3,5,6,

%U 2,2,1,4,4,5,3,3,1,2,5,4,5,5,2,4,5,7,2,5,1,5,4,4,4,6,3,2,6,4,5,4

%N Number of ways to write n = p + sum_{k=1..m}(-1)^(m-k)*p_k, where p is a Sophie Germain prime and p_k is the k-th prime.

%C Conjecture: a(n)>0 for all n>2.

%C This has been verified for n up to 10^7.

%C Let s_n=sum_{k=1}^n(-1)^{n-k}p_k for n=1,2,3,... The author also made the following conjectures:

%C (1) For each n>2, there is an integer k>0 such that 3(n-s_k)-1 and 3(n-s_k)+1 are twin primes.

%C (2) For each n>3, there is an integer k>0 such that 3(n-s_k)-2 and 3(n-s_k)+2 are cousin primes.

%C (3) Every n=6,7,... can be written as p+s_k (k>0) with p and p+6 sexy primes.

%C (4) Any integer n>3 different from 65 and 365 can be written as p+s_k (k>0) with p a term of A210479.

%C (5) Each integer n>8 can be written as q+s_k (k>0) with q-4, q, q+4 all practical.

%C (6) Any integer n>1 can be written as j(j+1)/2+s_k with j,k>0.

%H Zhi-Wei Sun, <a href="/A213202/b213202.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.

%e a(11)=1 since 11=3+p_5-p_4+p_3-p_2+p_1 with 3 and 2*3+1 both prime.

%e a(182)=1 since 182=179+(7-5+3-2) with 179 and 2*179+1 both prime.

%t sp[n_]:=qq[n]=PrimeQ[n]&&PrimeQ[2n+1]

%t s[0_]:=0

%t s[n_]:=s[n]=Prime[n]-s[n-1]

%t a[n_]:=a[n]=Sum[If[n-s[m]>0&&sp[n-s[m]],1,0],{m,1,n}]

%t Do[Print[n," ",a[n]],{n,1,100}]

%Y Cf. A005384, A008347, A005153, A210479.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Mar 01 2013