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%I #24 Feb 27 2020 22:18:44
%S 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,
%T 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,
%U 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
%N Number of ordered ways of writing n = i + j, where i is 0 or a prime and j is a triangular number (A000217) >= 0.
%C 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.
%C _Zhi-Wei Sun_ has offered a monetary reward for settling this conjecture.
%C No counterexample below 10^10. - _D. S. McNeil_
%C 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.
%C 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.
%C 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
%H T. D. Noe, <a href="/A132399/b132399.txt">Table of n, a(n) for n = 0..10000</a>
%H T. D. Noe, <a href="http://www.sspectra.com/math/A132399c.gif">Plot of A132399(n) for n to 10^6</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A1=ind0803&L=nmbrthry">Posing to Number Theory List (1)</a>
%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9ebd2e77.0803">Posting to Number Theory List (2)</a>
%H Zhi-Wei Sun, <a href="http://arXiv.org/abs/0803.3737">Conjectures on sums of primes and triangular numbers</a>, J. Combin. Number Theory 1 (2009) 65-76 and arXiv:0803.3737
%e 0 = 0+0, so a(0) = 1,
%e 3 = 3+0 = 2+1 = 0+3, so a(3) = 3.
%e 8 = 7+1 = 5+3 = 2+6, so a(8) = 3.
%Y Cf. A117054, A144590.
%Y 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).
%Y Cf. A154752 (smallest prime p for each n). - _T. D. Noe_, Jan 19 2009
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Mar 23 2008
%E Corrected, edited and extended by _T. D. Noe_, Mar 26 2008
%E Edited by _N. J. A. Sloane_, Jan 15 2009
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