A117054 -id:A117054 - cp's OEIS frontend

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

N. J. A. Sloane, Mar 23 2008

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

			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.

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

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

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

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

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.

Original entry on oeis.org

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

N. J. A. Sloane, Jan 15 2009

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).

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

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.

Cf. A132399. Bisection of A117054.

cp's OEIS Frontend

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.

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