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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. 5

%I #14 Feb 27 2020 16:48:41

%S 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,

%T 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,

%U 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

%N 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.

%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 for n >= 2.

%C _Zhi-Wei Sun_ has offered a monetary reward for settling this conjecture.

%C No counterexample exists below 10^10 (_D. S. McNeil_).

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

%H T. D. Noe, <a href="/A144590/b144590.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 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>, arXiv:0803.3737.

%Y Cf. A132399. Bisection of A117054.

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_, Jan 15 2009

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