cp's OEIS Frontend

A117054 Number of ordered ways of writing n = i + j, where i is a prime and j is of the form k*(k+1), k > 0.

%I A117054 #14 Feb 27 2020 22:17:15
%S A117054 0,0,0,1,1,0,1,1,2,0,1,0,2,1,2,0,2,0,3,0,1,1,3,0,4,0,1,0,2,0,3,1,3,0,
%T A117054 3,0,3,0,2,0,2,0,5,1,2,0,3,0,6,0,1,0,4,0,3,0,1,1,5,0,5,0,3,0,3,0,4,0,
%U A117054 2,0,3,0,7,1,3,0,3,0,6,0,2,0,4,0,6,0,2,0,4,0,5,1,3,0,5,0,3,0,3,0,5,0,8,0,1
%N A117054 Number of ordered ways of writing n = i + j, where i is a prime and j is of the form k*(k+1), k > 0.
%C A117054 Based on a posting by _Zhi-Wei Sun_ to the Number Theory Mailing List, Mar 23 2008, where he conjectures that a(2n+1) > 0 for n >= 2.
%C A117054 _Zhi-Wei Sun_ has offered a monetary reward for settling this conjecture.
%C A117054 No counterexample below 10^10. - _D. S. McNeil_
%H A117054 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 A117054 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 A117054 Zhi-Wei Sun, <a href="http://arXiv.org/abs/0803.3737">Conjectures on sums of primes and triangular numbers</a>, arxiv:0803.3737 and Journal of Combinatorics and Number Theory, 1 (2009), no.1, 65-76
%p A117054 t0:=array(0..300); for n from 0 to 300 do t0[n]:=0; od:
%p A117054 t1:=[seq(ithprime(i),i=1..70)]; t2:=[seq(n*(n+1),n=1..30)];
%p A117054 for i from 1 to 70 do for j from 1 to 30 do k:=t1[i]+t2[j]; if k <= 300 then t0[k]:=t0[k]+1; fi; od: od:
%p A117054 t3:=[seq(t0[n],n=1..300)];
%Y A117054 Cf. A132399, A144590.
%K A117054 nonn,easy
%O A117054 0,9
%A A117054 _N. J. A. Sloane_, Jan 15 2009