A238405 a(n) = |{0 < k < n: prime(k) + pi(n-k) is a triangular number}|, where pi(.) is given by A000720.

0, 0, 2, 0, 1, 0, 1, 2, 3, 3, 2, 2, 1, 3, 3, 1, 2, 1, 2, 5, 3, 3, 4, 1, 2, 2, 3, 3, 1, 2, 3, 4, 5, 6, 5, 3, 2, 2, 3, 3, 2, 5, 5, 4, 3, 2, 4, 3, 2, 3, 3, 2, 4, 6, 4, 6, 9, 8, 6, 4, 3, 2, 3, 4, 5, 3, 5, 6, 5, 5, 1, 1, 3, 5, 4, 4, 9, 7, 6, 6, 4, 6, 3, 3, 5, 8, 8, 5, 4, 7, 8, 4, 5, 3, 2, 3, 4, 4, 4, 4

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 6, and a(n) = 1 only for n = 5, 7, 13, 16, 18, 24, 29, 71, 72, 158.

Links

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

Example

a(7) = 1 since 7 = 2 + 5 with prime(2) + pi(5) = 3 + 3 = 3*4/2.
a(24) = 1 since 24 = 4 + 20 with prime(4) + pi(20) = 7 + 8 = 5*6/2.
a(158) = 1 since 158 = 148 + 10 with prime(148) + pi(10) = 857 + 4 = 41*42/2.

Mathematica

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

Crossrefs

Cf. A000040, A000217, A238386, A238402.

Sequence in context: A071483 A340499 A336033 * A004173 A185370 A352747

Adjacent sequences: A238402 A238403 A238404 * A238406 A238407 A238408

Keyword

nonn

Author

Zhi-Wei Sun, Feb 26 2014

Status

approved