A301461 Number of integers less than or equal to n whose largest prime factor is 3.

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

Offset

0,7

Comments

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

A distinct sequence can be generated for each prime number; this sequence is for the prime number 3. For an example using another prime number see A301506.

Links

Table of n, a(n) for n=0..78.

Formula

From David A. Corneth, Mar 27 2018 (Start)

a(n) - a(n - 1) = 1 if and only if n is in 3 * A003586. If n isn't in that sequence then a(n) = a(n - 1).

a(3 * n + b) = A071521(n), n > 0, 0 <= b < 3. (End)

Example

a(12) = a(2^2 * 3^1); 3 is the largest prime factor, so a(12) exceeds the previous term by 1. For a(13), 13 is a prime, so there is no increase from the previous term.

Mathematica

Accumulate@ Array[Boole[FactorInteger[#][[-1, 1]] == 3] &, 80, 0] (* Michael De Vlieger, Apr 21 2018 *)

Prog

(MATLAB)
clear; clc;
prime = 3;
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]'
(PARI) gpf(n) = if (n<=1, n, vecmax(factor(n)[, 1]));
a(n) = sum(k=1, n, gpf(k)==3); \\ Michel Marcus, Mar 27 2018

Crossrefs

Cf. A003586, A065119, A071521.

Cf. A301506.

Sequence in context: A329194 A210533 A246435 * A172471 A046155 A026819

Adjacent sequences: A301458 A301459 A301460 * A301462 A301463 A301464

Keyword

nonn

Author

Ralph-Joseph Tatt, Mar 21 2018

Status

approved