

A220413


Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime


15



1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 4, 1, 4, 2, 3, 3, 3, 5, 5, 5, 3, 3, 5, 4, 4, 5, 6, 7, 4, 4, 5, 2, 6, 5, 5, 5, 4, 2, 4, 6, 4, 5, 4, 4, 8, 6, 5, 11, 6, 6, 8, 10, 5, 5, 5, 8, 6, 6, 11, 7, 5, 7, 9, 7, 6, 7, 8, 9, 6, 8, 10, 7, 11, 8, 7, 10, 9, 9, 6, 5, 7, 8, 13, 7, 9, 13, 13, 12, 9, 9
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OFFSET

1,6


COMMENTS

Conjecture: a(n)>0 for every n=1,2,3,... Moreover, any integer n>3 not among 7, 22, 31 can be written as p+q (q>0) with p and p^3+2*q^3 both prime.
We have verified this conjecture for n up to 10^8. D. R. HeathBrown proved in 2001 that there are infinitely many primes in the form x^3+2*y^3, where x and y are positive integers.
ZhiWei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y (x>=0, y>=0) with x^m+2*y^m prime.
When m=1, this follows from Bertrand's postulate proved by Chebyshev in 1850. For m = 5, 7, 9, 11, 13, 15, 17, 19, it suffices to require that n is greater than 46, 69, 141, 274, 243, 189, 320, 454 respectively.


REFERENCES

D. R. HeathBrown, Primes represented by x^3+2*y^3, Acta Arith. 186(2001), 184.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.


EXAMPLE

a(9)=1 since 9=7+2 with 7^3+2*2^3=359 prime.
a(22)=1 since 22=1+21 with 1^3+2*21^3=18523 prime.


MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[k^3+2(nk)^3]==True, 1, 0], {k, 0, n}]
Do[Print[n, " ", a[n]], {n, 1, 100}]


CROSSREFS

Cf. A220272, A219842, A219864, A219923.
Sequence in context: A112224 A058774 A033101 * A029217 A161230 A161054
Adjacent sequences: A220410 A220411 A220412 * A220414 A220415 A220416


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 13 2012


STATUS

approved



