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A230110 Composite numbers m such that Sum_{i=1..k} (p_i/(p_i+1)) + Product_{i=1..k} (p_i/(p_i-1)) is an integer, where p_i are the k prime factors of m (with multiplicity).

%I A230110 #14 Mar 31 2023 05:16:02
%S A230110 8,10,30,63,64,512,588,720,800,1320,3960,4096,5184,5760,6400,7200,
%T A230110 21600,27720,27900,32768,35280,41472,46080,51200,70840,84672,92400,
%U A230110 95040,105600,151200,188160,262144,331776,368640,376320,409600,504000,518400,576000,640000
%N A230110 Composite numbers m such that Sum_{i=1..k} (p_i/(p_i+1)) + Product_{i=1..k} (p_i/(p_i-1)) is an integer, where p_i are the k prime factors of m (with multiplicity).
%C A230110 Includes 2^(3*a) * 3^(4*b) if 3*a >= 4*b. - _Robert Israel_, Mar 30 2023
%H A230110 Robert Israel, <a href="/A230110/b230110.txt">Table of n, a(n) for n = 1..155</a>
%e A230110 Prime factors of 3960 are 2^3, 3^2, 5 and 11.
%e A230110 Sum_{i=1..7} (p(i)/(p(i)+1)) = 3*(2/(2+1)) + 2*(3/(3+1)) + 5/(5+1) + 11/(11+1) = 21/4.
%e A230110 Product_{i=1..7} (p(i)/(p(i)-1)) = (2/(2+1))^3 * (3/(3-1))^2 * 5/(5-1) * 11/(11-1) = 99/4.
%e A230110 Their sum is an integer: 21/4 + 99/4 = 30.
%p A230110 with(numtheory); P:=proc(i) local b,d,n,p;
%p A230110 for n from 2 to i do p:=ifactors(n)[2];
%p A230110 b:=add(op(2,d)*op(1,d)/(op(1,d)+1),d=p)+mul((op(1,d)/(op(1,d)-1))^op(2,d),d=p);
%p A230110 if trunc(b)=b then print(n); fi; od; end: P(10^7);
%Y A230110 Cf. A199767, A198391, A227034, A227248, A230111, A230112.
%K A230110 nonn
%O A230110 1,1
%A A230110 _Paolo P. Lava_, Oct 09 2013