A213202 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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2013

Keywords

• nonn

Comments

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

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

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

(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.

(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.

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

(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.

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

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

Examples

			a(11)=1 since 11=3+p_5-p_4+p_3-p_2+p_1 with 3 and 2*3+1 both prime.
a(182)=1 since 182=179+(7-5+3-2) with 179 and 2*179+1 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Crossrefs

Cf. A005384, A008347, A005153, A210479.

Programs

Mathematica

sp[n_]:=qq[n]=PrimeQ[n]&&PrimeQ[2n+1]
s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
a[n_]:=a[n]=Sum[If[n-s[m]>0&&sp[n-s[m]],1,0],{m,1,n}]
Do[Print[n," ",a[n]],{n,1,100}]