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

0,11

Comments

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.

Links

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

Example

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.

Maple

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

Mathematica

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

Prog

(MATLAB)
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]'

Crossrefs

Cf. A080193.

Cf. A301461.

Sequence in context: A235402 A092278 A105512 * A002266 A075249 A008648

Adjacent sequences: A301503 A301504 A301505 * A301507 A301508 A301509

Keyword

nonn

Author

Ralph-Joseph Tatt, Mar 22 2018

Status

approved