A117054 Number of ordered ways of writing n = i + j, where i is a prime and j is of the form k*(k+1), k > 0.

0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 0, 2, 1, 2, 0, 2, 0, 3, 0, 1, 1, 3, 0, 4, 0, 1, 0, 2, 0, 3, 1, 3, 0, 3, 0, 3, 0, 2, 0, 2, 0, 5, 1, 2, 0, 3, 0, 6, 0, 1, 0, 4, 0, 3, 0, 1, 1, 5, 0, 5, 0, 3, 0, 3, 0, 4, 0, 2, 0, 3, 0, 7, 1, 3, 0, 3, 0, 6, 0, 2, 0, 4, 0, 6, 0, 2, 0, 4, 0, 5, 1, 3, 0, 5, 0, 3, 0, 3, 0, 5, 0, 8, 0, 1

Offset: 0

Author

N. J. A. Sloane, Jan 15 2009

Keywords

• nonn
• easy

Comments

Based on a posting by Zhi-Wei Sun to the Number Theory Mailing List, Mar 23 2008, where he conjectures that a(2n+1) > 0 for n >= 2.

Zhi-Wei Sun has offered a monetary reward for settling this conjecture.

No counterexample below 10^10. - D. S. McNeil

Links

• Zhi-Wei Sun, Posing to Number Theory List (1)
• Zhi-Wei Sun, Posting to Number Theory List (2)
• Zhi-Wei Sun, Conjectures on sums of primes and triangular numbers, arxiv:0803.3737 and Journal of Combinatorics and Number Theory, 1 (2009), no.1, 65-76

Crossrefs

Cf. A132399, A144590.

Programs

Maple

t0:=array(0..300); for n from 0 to 300 do t0[n]:=0; od:
t1:=[seq(ithprime(i),i=1..70)]; t2:=[seq(n*(n+1),n=1..30)];
for i from 1 to 70 do for j from 1 to 30 do k:=t1[i]+t2[j]; if k <= 300 then t0[k]:=t0[k]+1; fi; od: od:
t3:=[seq(t0[n],n=1..300)];