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 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. 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 (list; graph; refs; listen; history; text; internal format)
 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)^{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. 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. EXAMPLE 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. 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}] 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 Zhi-Wei Sun, Mar 01 2013 STATUS approved

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Last modified May 18 09:26 EDT 2021. Contains 343995 sequences. (Running on oeis4.)