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.

%I A132399 #24 Feb 27 2020 22:18:44
%S A132399 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 A132399 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 A132399 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 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.
%C A132399 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 A132399 _Zhi-Wei Sun_ has offered a monetary reward for settling this conjecture.
%C A132399 No counterexample below 10^10. - _D. S. McNeil_
%C A132399 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 A132399 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 A132399 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 A132399 T. D. Noe, <a href="/A132399/b132399.txt">Table of n, a(n) for n = 0..10000</a>
%H A132399 T. D. Noe, <a href="http://www.sspectra.com/math/A132399c.gif">Plot of A132399(n) for n to 10^6</a>
%H A132399 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A1=ind0803&amp;L=nmbrthry">Posing to Number Theory List (1)</a>
%H A132399 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 A132399 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 A132399 0 = 0+0, so a(0) = 1,
%e A132399 3 = 3+0 = 2+1 = 0+3, so a(3) = 3.
%e A132399 8 = 7+1 = 5+3 = 2+6, so a(8) = 3.
%Y A132399 Cf. A117054, A144590.
%Y A132399 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 A132399 Cf. A154752 (smallest prime p for each n). - _T. D. Noe_, Jan 19 2009
%K A132399 nonn,easy
%O A132399 0,4
%A A132399 _N. J. A. Sloane_, Mar 23 2008
%E A132399 Corrected, edited and extended by _T. D. Noe_, Mar 26 2008
%E A132399 Edited by _N. J. A. Sloane_, Jan 15 2009