A132399 Number of ordered ways of writing n = i + j, where i is 0 or a prime and j is a triangular number (A000217) >= 0.

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

Offset: 0

Author

N. J. A. Sloane, Mar 23 2008

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 except for n = 216.

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

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

Note that A076768 contains 216 and the numbers n whose only representation has 0 instead of a prime; all other integers appear to be the sum of a prime and a triangular number. Except for n=216, there is no other n < 2*10^9 for which a(n)=0.

It is clear that a(t) > 0 for any triangular number t because we always have the representation t = t+0. Triangular numbers tend to have only a few representations. Hence by not plotting a(n) for triangular n, the plot (see link) more clearly shows how a(n) slowly increases as n increases. This is more evidence that 216 is the only exception.

216 is the only exception less than 10^12. Let p(n) be the least prime (or 0 if n is triangular) such that n = p(n) + t(n), where t(n) is a triangular number. For n < 10^12, the largest value of p(n) is only 2297990273, which occurs at n=882560134401. - T. D. Noe, Jan 23 2009

Examples

			0 = 0+0, so a(0) = 1,
3 = 3+0 = 2+1 = 0+3, so a(3) = 3.
8 = 7+1 = 5+3 = 2+6, so a(8) = 3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• T. D. Noe, Plot of A132399(n) for n to 10^6
• 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, J. Combin. Number Theory 1 (2009) 65-76 and arXiv:0803.3737

Crossrefs

Cf. A117054, A144590.

Cf. A065397 (primes p whose only representation as the sum of a prime and a triangular number is p+0), A090302 (largest prime p for each n).

Cf. A154752 (smallest prime p for each n). - T. D. Noe, Jan 19 2009

Extensions

Corrected, edited and extended by T. D. Noe, Mar 26 2008

Edited by N. J. A. Sloane, Jan 15 2009