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a(n) is the number of products P of distinct positive integers satisfying P <= n.
1

%I #15 Dec 09 2020 16:04:35

%S 0,1,2,3,4,7,8,11,12,15,16,21,22,25,28,31,32,37,38,43,46,49,50,59,60,

%T 63,66,71,72,81,82,87,90,93,96,105,106,109,112,121,122,131,132,137,

%U 142,145,146,159,160,165,168,173,174,183,186,195

%N a(n) is the number of products P of distinct positive integers satisfying P <= n.

%C The products must contain at least two integers. - _Sean A. Irvine_, Oct 13 2019

%H Robert Israel, <a href="/A026808/b026808.txt">Table of n, a(n) for n = 1..10000</a>

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a026/A026808.java">Java program</a> (github)

%e a(6) = 7 because the possible products are 1*2, 1*3, 1*4, 1*5, 1*6, 2*3, and 1*2*3. - _Sean A. Irvine_, Oct 13 2019

%p N:= 200: # for a(1)..a(N)

%p with(queue):Q:= new():

%p V:= Vector(N):

%p for i from 1 to N/2 do

%p for j from i+1 to N while i*j <= N do

%p p:= i*j;

%p V[p]:= V[p]+1;

%p if p*(j+1) <= N then enqueue(Q,[i,j]) fi

%p od od:

%p while not empty(Q) do

%p s:= dequeue(Q);

%p p:= convert(s,`*`);

%p for k from s[-1]+1 to N while p*k <= N do

%p pp:= p*k;

%p V[pp]:= V[pp]+1;

%p if pp*(k+1) <= N then enqueue(Q,[op(s),k]) fi

%p od;

%p od:

%p ListTools:-PartialSums(convert(V,list)); # _Robert Israel_, Dec 09 2020

%Y Partial sums of A254578.

%K nonn

%O 1,3

%A _Clark Kimberling_