A112751 Number of numbers of the form 3^i*5^j that are less than or equal to n.

1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset

1,3

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

From Ridouane Oudra, Oct 26 2025: (Start)

a(n) = Card_{ k | A003593(k) <= n }.

a(n) = Sum_{k=1..n} mu(15*k)*floor(n/k).

a(n) = Sum_{k=1..n} (floor(15^k/k)-floor((15^k-1)/k)).

a(n) = Sum_{i=0..floor(log_3(n))} (floor(log_5(n/3^i)) + 1).

Asymptotically: a(n) ~ c*log(3*n)*log(5*n), where c=1/(2*log(3)*log(5)). (End)

Maple

with(numtheory): seq(add(mobius(15*k)*floor(n/k), k=1..n), n=1..100); # Ridouane Oudra, Oct 26 2025

Mathematica

Accumulate[Table[Boole[n == Times @@ ({3, 5}^IntegerExponent[n, {3, 5}])], {n, 1, 100}]] (* Amiram Eldar, May 04 2025 *)

Prog

(Magma) [&+[MoebiusMu(15*k)*Floor(n/k):k in [1..n]]: n in [1..97]]; // Marius A. Burtea, Jul 30 2019
(Python)
from sympy import integer_log
def A112751(n): return sum(integer_log(n//5**i, 3)[0]+1 for i in range(integer_log(n, 5)[0]+1)) # Chai Wah Wu, Jan 28 2026

Crossrefs

Cf. A003593, A008683, A071520.

Sequence in context: A334097 A122027 A359121 * A316843 A331246 A331256

Adjacent sequences: A112748 A112749 A112750 * A112752 A112753 A112754

Keyword

nonn

Author

Reinhard Zumkeller, Sep 18 2005

Status

approved