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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).
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%I #14 Mar 31 2023 05:16:02

%S 8,10,30,63,64,512,588,720,800,1320,3960,4096,5184,5760,6400,7200,

%T 21600,27720,27900,32768,35280,41472,46080,51200,70840,84672,92400,

%U 95040,105600,151200,188160,262144,331776,368640,376320,409600,504000,518400,576000,640000

%N 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 Includes 2^(3*a) * 3^(4*b) if 3*a >= 4*b. - _Robert Israel_, Mar 30 2023

%H Robert Israel, <a href="/A230110/b230110.txt">Table of n, a(n) for n = 1..155</a>

%e Prime factors of 3960 are 2^3, 3^2, 5 and 11.

%e 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 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 Their sum is an integer: 21/4 + 99/4 = 30.

%p with(numtheory); P:=proc(i) local b,d,n,p;

%p for n from 2 to i do p:=ifactors(n)[2];

%p 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 if trunc(b)=b then print(n); fi; od; end: P(10^7);

%Y Cf. A199767, A198391, A227034, A227248, A230111, A230112.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Oct 09 2013