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%I #32 Sep 23 2020 02:48:13
%S 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,
%T 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,
%U 10,10,10,10,10,10,10,10,10,10,11,11,11,11,11
%N Number of integers less than or equal to n whose largest prime factor is 5.
%C a(n) increases when n has the form 2^a*3^b*5^c, with a,b >= 0 and c > 0.
%C 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.
%H Robert Israel, <a href="/A301506/b301506.txt">Table of n, a(n) for n = 0..10000</a>
%e 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.
%p N:= 100: # for a(0)..a(N)
%p L:= sort([seq(seq(seq(2^a*3^b*5^c, c=1..floor(log[5](N/(2^a*3^b)))),
%p b = 0..floor(log[3](N/2^a))), a = 0 .. floor(log[2](N)))]):
%p V:= Array(0..N):
%p V[L]:= 1:
%p ListTools:-PartialSums(convert(V,list)); # _Robert Israel_, Sep 22 2020
%t Accumulate@ Array[Boole[FactorInteger[#][[-1, 1]] == 5] &, 80, 0] (* _Michael De Vlieger_, Apr 21 2018 *)
%o (MATLAB)
%o clear;clc;
%o prime = 5;
%o limit = 10000;
%o largest_divisor = ones(1,limit+1);
%o for k = 0:limit
%o f = factor(k);
%o largest_divisor(k+1) = f(end);
%o end
%o for i = 1:limit+1
%o FQN(i) = sum(largest_divisor(1:i)==prime);
%o end
%o output = [0:limit;FQN]'
%Y Cf. A080193.
%Y Cf. A301461.
%K nonn
%O 0,11
%A _Ralph-Joseph Tatt_, Mar 22 2018
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