A301461 -id:A301461 - cp's OEIS frontend

A301506 Number of integers less than or equal to n whose largest prime factor is 5.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11

Offset: 0

Ralph-Joseph Tatt, Mar 22 2018

nonn

a(n) increases when n has the form 2^a*3^b*5^c, with a,b >= 0 and c > 0.

A distinct sequence can be generated for each prime number; this sequence is for the prime number 5. For an example using another prime number see A301461.

			a(15) = a(2^0 * 3^1 * 5^1); 5 is the largest prime factor, so a(15) exceeds the previous term by 1. For a(16) = a(2^4), there is no increase from the previous term.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A080193.

Cf. A301461.

clear;clc;
prime = 5;
limit = 10000;
largest_divisor = ones(1,limit+1);
for k = 0:limit
    f = factor(k);
    largest_divisor(k+1) = f(end);
end
for i = 1:limit+1
    FQN(i) = sum(largest_divisor(1:i)==prime);
end
output = [0:limit;FQN]'

N:= 100: # for a(0)..a(N)
L:= sort([seq(seq(seq(2^a*3^b*5^c, c=1..floor(log[5](N/(2^a*3^b)))),
  b = 0..floor(log[3](N/2^a))), a = 0 .. floor(log[2](N)))]):
V:= Array(0..N):
V[L]:= 1:
ListTools:-PartialSums(convert(V,list)); # Robert Israel, Sep 22 2020

Accumulate@ Array[Boole[FactorInteger[#][[-1, 1]] == 5] &, 80, 0] (* Michael De Vlieger, Apr 21 2018 *)

A382487 The number of divisors of n whose largest prime factor is 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 5, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 3, 0, 0, 1

Offset: 1

Amiram Eldar, Mar 29 2025

The number of 3-smooth divisors of n that are not powers of 2.

The number of terms of A065119 that divide n.

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

Cf. A001511, A001651, A065119, A072078, A007949, A051064, A169611, A301461, A306771.

a[n_] := (IntegerExponent[n, 2] + 1) * IntegerExponent[n, 3]; Array[a, 100]

a(n) = (valuation(n, 2) + 1) * valuation(n, 3);

a(n) = A072078(n) - A001511(n).
a(n) = A001511(n) * A007949(n).
a(n) = 0 if and only if n is in A001651.
a(n) = 1 if and only if n is in A306771.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.
In general, the asymptotic mean of the number prime(k+1)-smooth divisors of n that are not prime(k)-smooth, for k >= 1, is (1/(prime(k+1)-1)) * Product_{i=1..k} (prime(i)/(prime(i)-1)).
Dirichlet g.f.: (zeta(s)/(1-1/2^s))*(1/(1-1/3^s) - 1).

cp's OEIS Frontend

A301506 Number of integers less than or equal to n whose largest prime factor is 5.

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