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A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

%I A238922 #25 Jan 10 2025 04:29:52
%S A238922 1,12,18,220,396,17296,24016,287532,4661056,64288512,334144656,
%T A238922 358585488,555192576,568719616,2172649216,2451538112,2645953344,
%U A238922 2955423888,6704333824,26996772032,88734733632,147861504000,311063879024,371226582848,429391876096
%N A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.
%C A238922 A212128 and A230164 are subsets of this sequence.
%C A238922 a(26) > 10^12. - _Giovanni Resta_, Mar 11 2014
%e A238922 Divisors of 12 are 1, 2, 3, 4, 6, 12 and 1/1 + 1/2 + 1/3 +1/4 + 1/6 + 1/12 = 7/3. Prime factors of 12 are 2^2, 3 and 1/2 + 1/2 + 1/3 = 4/3. Finally 7/3 - 4/3 = 1 that is an integer.
%p A238922 with(numtheory); P:=proc(q) local a,b,c,k,n;
%p A238922 for n from 1 to q do if not isprime(n) then b:=sigma(n)/n;
%p A238922 a:=ifactors(n)[2]; c:=add(a[k][2]/a[k][1],k=1..nops(a));
%p A238922 if type(b-c,integer) then lprint(n,b-c); fi; fi; od; end: P(10^6);
%Y A238922 Cf. A003415, A212128, A224346, A230164.
%K A238922 nonn,hard
%O A238922 1,2
%A A238922 _Paolo P. Lava_, Mar 07 2014
%E A238922 a(9)-a(10), a(13)-a(17), a(19)-a(25) from _Giovanni Resta_, Mar 11 2014