A125072 - OEIS

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

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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). - 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

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) = 2.

MATHEMATICA

f[n_] := Length @ 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.

A125072(n) = vecsum(apply(e -> A010054(e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 08 2017

CROSSREFS

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

Sequence in context: A029369 A255315 A373029 * A162642 A366246 A361205

Adjacent sequences: A125069 A125070 A125071 * A125073 A125074 A125075

KEYWORD

nonn,easy

AUTHOR

Leroy Quet, Nov 18 2006

EXTENSIONS

Extended by Ray Chandler, Nov 19 2006

STATUS

approved