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A220554 Number of ways to write 2n = p+q (q>0) with p, 2p+1 and (p-1)^2+q^2 all prime 5
0, 2, 3, 2, 2, 2, 2, 3, 3, 3, 1, 1, 2, 3, 3, 1, 2, 3, 4, 3, 4, 2, 2, 2, 3, 1, 3, 3, 5, 3, 1, 2, 2, 2, 5, 2, 1, 2, 2, 5, 1, 2, 4, 3, 4, 4, 3, 5, 4, 4, 1, 2, 2, 2, 4, 4, 4, 4, 6, 6, 4, 2, 6, 4, 4, 4, 2, 2, 5, 6, 3, 2, 3, 5, 5, 4, 3, 2, 4, 4, 2, 4, 4, 4, 4, 3, 4, 3, 5, 6, 3, 4, 5, 5, 3, 1, 2, 5, 3, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 2*10^8. It implies that there are infinitely many Sophie Germain primes.

Note that Ming-Zhi Zhang asked (before 1990) whether any odd integer greater than 1 can be written as x+y (x,y>0) with x^2+y^2 prime, see A036468.

Zhi-Wei Sun also made the following related conjectures:

(1) Any integer n>2 can be written as x+y (x,y>=0) with 3x-1, 3x+1 and x^2+y^2-3(n-1 mod 2) all prime.

(2) Each integer n>3 not among 20, 40, 270 can be written as x+y (x,y>0) with 3x-2, 3x+2 and x^2+y^2-3(n-1 mod 2) all prime.

(3) Any integer n>4 can be written as x+y (x,y>0) with 2x-3, 2x+3 and x^2+y^2-3(n-1 mod 2) all prime. Also, every n=10,11,... can be written as x+y (x,y>=0) with x-3, x+3 and x^2+y^2-3(n-1 mod 2) all prime.

(4) Any integer n>97 can be written as p+q (q>0) with p, 2p+1, n^2+pq all prime. Also, each integer n>10 can be written as p+q (q>0) with p, p+6, n^2+pq all prime.

(5) Every integer n>3 different from 8 and 18 can be written as x+y (x>0, y>0) with 3x-2, 3x+2 and n^2-xy all prime.

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 2nd Edition, Springer, New York, 2004, p. 161.

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.

EXAMPLE

a(16)=1 since 32=11+21 with 11, 2*11+1=23 and (11-1)^2+21^2=541 all prime.

MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[p]==True&&PrimeQ[2p+1]==True&&PrimeQ[(p-1)^2+(2n-p)^2]==True, 1, 0], {p, 1, 2n-1}]

Do[Print[n, " ", a[n]], {n, 1, 1000}]

CROSSREFS

Cf. A005384, A005385, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

Sequence in context: A031217 A064131 A111497 * A208243 A209320 A097051

Adjacent sequences:  A220551 A220552 A220553 * A220555 A220556 A220557

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 15 2012

STATUS

approved

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Last modified October 21 16:50 EDT 2019. Contains 328302 sequences. (Running on oeis4.)