A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime

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

Offset

1,2

Comments

Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime.

This has been verified for n up to 2*10^6.

Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime.

Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(3)=1 since 2*3-1=5=1+4 with 1^18+3*4^18=206158430209 prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[k^18+3*(2n-1-k)^18]==True, 1, 0], {k, 0, 2n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A220554, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

Sequence in context: A286361 A357849 A097295 * A083896 A332083 A084115

Adjacent sequences: A220569 A220570 A220571 * A220573 A220574 A220575

Keyword

nonn

Author

Zhi-Wei Sun, Dec 16 2012

Status

approved