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
Keywords
Examples
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.
Programs
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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]' -
Mathematica
Accumulate@ Array[Boole[FactorInteger[#][[-1, 1]] == 3] &, 80, 0] (* Michael De Vlieger, Apr 21 2018 *)
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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
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)
Comments