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.

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, 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, 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

Offset: 1

Author

T. D. Noe, Jan 17 2009

Keywords

• sign

Comments

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.

Examples

			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.
		

References

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

Links

• T. D. Noe, Table of n, a(n) for n=1..10001
• Zhi-Wei Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009.

Programs

Mathematica

nn=300; s=0; tri=Rest[Reap[i=0; While[s1 && !PrimeQ[p], m-- ]; If[m==1 && !PrimeQ[p], -1, p]], {n,nn}]

Extensions

Offset in b-file corrected by N. J. A. Sloane, Aug 31 2009