A224346 - OEIS

%I #21 Jan 09 2025 02:17:40

%S 1,2,21,44,560,752,2064,12224,98595,38735300,53668332,147728896,

%T 407729196,423212608,516441712,1227777925,1323319996,20440128681,

%U 153088685248,206158168064,375868306368,798666196041

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

%C If (Sum_{i=1..k} 1/p(i)) * (Sum_{i=1..j} 1/d(i)) is considered, for n between 1 and 10^6, only 1080 gives an integer value: 27/10 * 10/3 = 9.

%C a(23) > 10^12. - _Giovanni Resta_, Apr 10 2013

%e n=44; its prime factors are 2^2, 11 while its divisors are 1, 2, 4, 11, 22, 44 and 1/2 + 1/2 + 1/11 + 1/1 + 1/2 + 1/4 + 1/11 + 1/22 + 1/44 = 3.

%p with(numtheory); List224346:=proc(q) local a,b,c,j,n;

%p for n from 1 to q do

%p a:=ifactors(n)[2]; b:=0; for j from 1 to nops(a) do b:=b+a[j,2]/a[j,1]; od;

%p c:=sigma(n)/n; if type(b+c,integer) then print(n); fi; od; end:

%p List224346(10^6);

%t Select[Range[10^5], Mod[DivisorSigma[1,#] + Total[# / Divide @@@ FactorInteger@#], #] == 0 &] (* _Giovanni Resta_, Apr 10 2013 *)

%Y Cf. A000203, A001414.

%K nonn,more

%O 1,2

%A _Paolo P. Lava_, Apr 08 2013

%E a(1)=1 and a(10)-a(22) from _Giovanni Resta_, Apr 10 2013