OFFSET
1,2
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.
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.
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
sign
AUTHOR
T. D. Noe, Jan 17 2009
EXTENSIONS
Offset in b-file corrected by N. J. A. Sloane, Aug 31 2009
STATUS
approved