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A187785
Number of ways to write n=x+y (x,y>=0) with {6x-1,6x+1} a twin prime pair and y a triangular number.
2
1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 4, 0, 2, 2, 3, 4, 1, 3, 1, 3, 3, 3, 2, 3, 2, 3, 2, 2, 4, 2, 7, 1, 3, 2, 1, 6, 4, 4, 3, 1, 3, 2, 3, 6, 3, 6, 0, 3, 3, 2, 6, 2, 4, 1, 3, 4, 3, 3, 4, 4, 1, 1, 1, 3, 3, 6, 2, 2, 2, 2, 7, 1, 3, 3, 2, 5, 2, 5, 2, 1, 5, 1, 4, 1, 4, 4, 1, 3, 2, 3, 4, 2, 3, 4, 2, 5, 1, 3
OFFSET
1,2
COMMENTS
Conjecture: a(n)>0 for all n>48624 not equal to 76106.
We have verified this for n up to 2*10^8. It seems that 723662 is the unique n>76106 which really needs y=0 in the described representation.
Compare the conjecture with another Sun's conjecture associated with A132399.
REFERENCES
Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), no. 1, 65-76.
LINKS
Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(9)=1 since 9=3+3(3+1)/2 with 6*3-1 and 6*3+1 both prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[6(n-k(k+1)/2)-1]==True&&PrimeQ[6(n-k(k+1)/2)+1]==True, 1, 0], {k, 0, (Sqrt[8n+1]-1)/2}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 06 2013
STATUS
approved