A144590 Number of ordered ways of writing 2n+1 = i + j, where i is a prime and j is of the form k*(k+1), k > 0.

0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 1, 3, 4, 1, 2, 3, 3, 3, 3, 2, 2, 5, 2, 3, 6, 1, 4, 3, 1, 5, 5, 3, 3, 4, 2, 3, 7, 3, 3, 6, 2, 4, 6, 2, 4, 5, 3, 5, 3, 3, 5, 8, 1, 2, 9, 1, 7, 7, 3, 5, 5, 3, 3, 5, 4, 4, 7, 2, 4, 8, 2, 7, 5, 2, 4, 8, 3, 4, 6, 4, 6, 7, 2, 2, 12, 2, 6, 5, 2, 8, 5, 4, 6, 7, 2, 4, 11, 3, 4, 10, 3, 7, 6, 2

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(n) > 0 for n >= 2.

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

No counterexample exists below 10^10 (D. S. McNeil).

References

• Zhi-Wei Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory, 1(2009), no.1, 65-76.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Posing to Number Theory List (1)
• Zhi-Wei Sun, Posting to Number Theory List (2)
• Zhi-Wei Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009.

Crossrefs

Cf. A132399. Bisection of A117054.