A125073 a(n) = sum of the exponents in the prime factorization of n which are triangular numbers.

0, 1, 1, 0, 1, 2, 1, 3, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 4, 0, 2, 3, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 4, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 1, 6, 2, 3, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 4, 3

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Formula

Additive with a(p^e) = A010054(e)*e. - Antti Karttunen, Jul 08 2017

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=2} (k*(k+1)/2) * (P(k*(k+1)/2) - P(k*(k+1)/2 + 1)) = -0.10099019472003733178..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 28 2023

Example

The prime factorization of 360 is 2^3 *3^2 *5^1. There are two exponents in this factorization which are triangular numbers, 1 and 3. So a(360) = 1 + 3 = 4.

Mathematica

f[n_] := Plus @@ Select[Last /@ FactorInteger[n], IntegerQ[Sqrt[8# + 1]] &]; Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)

Prog

(PARI)
A010054(n) = issquare(8*n + 1); \\ This function from Michael Somos, Apr 27 2000.
A125073(n) = vecsum(apply(e -> (A010054(e)*e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 08 2017

Crossrefs

Cf. A010054, A077761, A125072, A125030, A125071.

Sequence in context: A147654 A321377 A071467 * A350387 A325310 A308881

Adjacent sequences: A125070 A125071 A125072 * A125074 A125075 A125076

Keyword

nonn,easy

Author

Leroy Quet, Nov 18 2006

Extensions

Extended by Ray Chandler, Nov 19 2006

Status

approved