A125072 a(n) = number of exponents in the prime-factorization of n which are triangular numbers.

0, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 2, 3

Offset: 1

Author

Leroy Quet, Nov 18 2006

Keywords

• nonn
• easy

Examples

			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) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A010054, A077761, A125073, A125029, A125070.

Programs

Mathematica

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

PARI

A010054(n) = issquare(8*n + 1); \\ This function from Michael Somos, Apr 27 2000.
A125072(n) = vecsum(apply(e -> A010054(e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 08 2017

Formula

Additive with a(p^e) = A010054(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} (P(k*(k+1)/2) - P(k*(k+1)/2 + 1)) = -0.34517646457715166126..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 28 2023

Extensions

Extended by Ray Chandler, Nov 19 2006