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A226365 Composite numbers such that Sum_{i=1..k} (1 + 1/p_i) - Product_{i=1..k} (1 + 1/p_i) is an integer, where p_i are the k prime factors of n (with multiplicity).

%I A226365 #16 Feb 24 2022 02:54:57
%S A226365 152,432,1620,1728,3456,4752,22464,46656,80892,139968,186624,237168,
%T A226365 326592,746496,1651968,2052864,2426112,2985984,5971968,10257408,
%U A226365 12177216,12690432,14048240,14183424,20155392,20901888,26127360,38817792
%N A226365 Composite numbers such that Sum_{i=1..k} (1 + 1/p_i) - Product_{i=1..k} (1 + 1/p_i) is an integer, where p_i are the k prime factors of n (with multiplicity).
%C A226365 The numbers of the sequence are the solution of the differential equation n' = (a-k)*n + b, which can also be written as A003415(n) = (a-k)*n + A003958(n), where k is the number of prime factors of n, and a is the integer Sum_{i=1..k} (1 + 1/p_i) - Product_{1=1..k} (1 + 1/p_i).
%C A226365 The numbers of the sequence satisfy also Sum_{i=1..k} (1 - 1/p_i) + Product_{i=1..k} (1 + 1/p_i) = some integer.
%H A226365 J. M. Borwein and E. Wong, <a href="http://www.math.ubc.ca/~erick/print/giuga25.pdf">A survey of results relating to Giuga's conjecture on primality</a>, May 8, 1995.
%H A226365 R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
%e A226365 237168 has prime factors 2, 2, 2, 2, 3, 3, 3, 3, 3, 61. 4*(1 + 1/2) + 5*(1 + 1/3) + (1 + 1/61) = 2504/183 is the sum over the 1 + 1/p_i. (1 + 1/2)^4 * (1 + 1/3)^5 * (1 + 1/61) = 3968/183 is the product of the 1 + 1/p_i. The difference over sum and product is 2504/183 - 3968/183 = -8, an integer.
%p A226365 with(numtheory); ListA226365:=proc(q) local a,d,n,p;
%p A226365 for n from 1 to q do if not isprime(n) then p:=ifactors(n)[2];
%p A226365 a:=add(op(2,d)+op(2,d)/op(1,d),d=p)-mul((1+1/op(1,d))^op(2,d),d=p);
%p A226365 if type(a,integer) then print(n); fi; fi;
%p A226365 od; end: ListA226365(10^9);
%Y A226365 Cf. A003415, A003958, A198391, A199767, A224912.
%K A226365 nonn
%O A226365 1,1
%A A226365 _Paolo P. Lava_, Jun 12 2013