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

Author

Reinhard Zumkeller, Sep 18 2005

Keywords

• nonn

Links

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

Crossrefs

Cf. A003593, A008683, A071520.

Programs

Magma

[&+[MoebiusMu(15*k)*Floor(n/k):k in [1..n]]: n in [1..97]]; // Marius A. Burtea, Jul 30 2019

Maple

with(numtheory): seq(add(mobius(15*k)*floor(n/k), k=1..n), n=1..90); # Ridouane Oudra, Jul 29 2019

Mathematica

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

Formula

From Ridouane Oudra, Jul 29 2019: (Start)

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

a(n) = Sum_{k=1..n} mu(15*k)*floor(n/k), where mu is the Möbius function (A008683).

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

From Ridouane Oudra, Jul 17 2020: (Start)

a(n) = Sum_{i=0..floor(log_5(n))} (floor(log_3(n/5^i)) + 1).

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