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Number of numbers of the form 3^i*5^j that are less than or equal to n.
2

%I #34 Mar 27 2023 18:31:17

%S 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,

%T 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,

%U 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

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

%F From _Ridouane Oudra_, Jul 29 2019: (Start)

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

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

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

%F From _Ridouane Oudra_, Jul 17 2020: (Start)

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

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

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

%o (Magma) [&+[MoebiusMu(15*k)*Floor(n/k):k in [1..n]]: n in [1..97]]; // _Marius A. Burtea_, Jul 30 2019

%Y Cf. A003593, A071520.

%K nonn

%O 1,3

%A _Reinhard Zumkeller_, Sep 18 2005