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A301461
Number of integers less than or equal to n whose largest prime factor is 3.
1
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.
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
KEYWORD
nonn
AUTHOR
Ralph-Joseph Tatt, Mar 21 2018
STATUS
approved