A235614 Number of ordered ways to write n = k + m with k a term of A235592 and m a positive triangular number.

0, 0, 1, 1, 2, 2, 2, 2, 3, 2, 2, 4, 1, 3, 3, 2, 3, 3, 3, 3, 5, 2, 3, 5, 3, 3, 3, 2, 4, 6, 2, 4, 3, 2, 4, 4, 4, 2, 6, 4, 4, 6, 2, 5, 2, 3, 7, 5, 4, 4, 6, 1, 2, 6, 5, 4, 5, 4, 5, 5, 1, 4, 7, 5, 5, 4, 2, 3, 5, 4, 4, 8, 4, 6, 4, 4, 4, 1, 2, 4, 7, 5, 3, 5, 3, 5, 3, 2, 6, 6, 4, 6, 8, 1, 4, 5, 5, 4, 7, 6

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 2.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(13) = 1 since 13 = 3 + 10 with 3*4 - prime(3) = 7 prime and 10 = 4*5/2 a positive triangular number.
a(52) = 1 since 52 = 37 + 15 with 37*38 - prime(37) = 1249 prime and 15 = 5*6/2 a positive triangular number.
a(61) = 1 since 61 = 6 + 55 with 6*7 - prime(6) = 29 prime and 55 = 10*11/2 a positive triangular number.
a(313) = 1 since 313 = 37 + 276 with 37*38 - prime(37) = 1249 prime and 276 = 23*24/2 a positive triangular number.

Mathematica

PQ[n_]:=PrimeQ[n(n+1)-Prime[n]]
TQ[n_]:=IntegerQ[Sqrt[8n+1]]
a[n_]:=Sum[If[PQ[k]&&TQ[n-k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A000217, A235592, A235613.

Sequence in context: A134868 A322861 A237259 * A127417 A308622 A198897

Adjacent sequences: A235611 A235612 A235613 * A235615 A235616 A235617

Keyword

nonn

Author

Zhi-Wei Sun, Jan 13 2014

Status

approved