cp's OEIS Frontend

A154752 Least prime p (or 0) such that n = p + T, where T is a triangular number (A000217), or -1 if there is no such representation.

%I A154752 #10 Sep 23 2015 13:07:05
%S A154752 0,2,0,3,2,0,7,2,3,0,5,2,3,11,0,13,2,3,13,5,0,7,2,3,19,5,17,0,19,2,3,
%T A154752 11,5,13,7,0,31,2,3,19,5,41,7,23,0,31,2,3,13,5,23,7,17,53,0,11,2,3,23,
%U A154752 5,61,7,53,19,29,0,31,2,3,67,5,17,7,19,47,31,11,0,13,2,3,37,5,29,7,31,59,43
%N A154752 Least prime p (or 0) such that n = p + T, where T is a triangular number (A000217), or -1 if there is no such representation.
%C A154752 Zhi-Wei Sun conjectures that only n=216 has no such representation. It appears that n = 2, 7, 61 and 211 are the only numbers for which the triangular number 0 is required in the representation (see A065397). When n is a triangular number, then a(n)=0. Sequence A132399 gives the number of representations of n as p+T. As n becomes larger, the largest prime required to verify the conjecture increases slowly. For example, for n<=10^3, the largest prime required is 953; for n<=10^6 it is 373361; for n<=10^9 it is only 36455351. Using primes less than 10^9, all n<243277591560 have been verified.
%D A154752 Zhi-Wei Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory, 1(2009), no.1, 65-76.
%H A154752 T. D. Noe, <a href="/A154752/b154752.txt">Table of n, a(n) for n=1..10001</a>
%H A154752 Zhi-Wei Sun, <a href="http://arXiv.org/abs/0803.3737">On sums of primes and triangular numbers</a>, arXiv:0803.3737 [math.NT], 2008-2009.
%e A154752 n=p+T: 1=0+1; 2=2+0; 3=0+3; 4=3+1; 5=2+3; 6=0+6; 7=7+0; 8=2+6; 9=3+6; 10=0+10.
%t A154752 nn=300; s=0; tri=Rest[Reap[i=0; While[s<nn, Sow[s]; i++; s=s+i]; Sow[s]]][[1,1]]; k=1; Table[If[n==tri[[k+1]], k++; 0, m=k; While[p=n-tri[[m]]; m>1 && !PrimeQ[p], m-- ]; If[m==1 && !PrimeQ[p], -1, p]], {n,nn}]
%K A154752 sign
%O A154752 1,2
%A A154752 _T. D. Noe_, Jan 17 2009
%E A154752 Offset in b-file corrected by _N. J. A. Sloane_, Aug 31 2009