

A213202


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


2



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
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OFFSET

1,4


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)^{nk}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(ns_k)1 and 3(ns_k)+1 are twin primes.
(2) For each n>3, there is an integer k>0 such that 3(ns_k)2 and 3(ns_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 q4, q, q+4 all practical.
(6) Any integer n>1 can be written as j(j+1)/2+s_k with j,k>0.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
ZhiWei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 27942812.


EXAMPLE

a(11)=1 since 11=3+p_5p_4+p_3p_2+p_1 with 3 and 2*3+1 both prime.
a(182)=1 since 182=179+(75+32) with 179 and 2*179+1 both prime.


MATHEMATICA

sp[n_]:=qq[n]=PrimeQ[n]&&PrimeQ[2n+1]
s[0_]:=0
s[n_]:=s[n]=Prime[n]s[n1]
a[n_]:=a[n]=Sum[If[ns[m]>0&&sp[ns[m]], 1, 0], {m, 1, n}]
Do[Print[n, " ", a[n]], {n, 1, 100}]


CROSSREFS

Cf. A005384, A008347, A005153, A210479.
Sequence in context: A165299 A071820 A055092 * A234972 A130326 A059906
Adjacent sequences: A213199 A213200 A213201 * A213203 A213204 A213205


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 01 2013


STATUS

approved



