A224346 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.

1, 2, 21, 44, 560, 752, 2064, 12224, 98595, 38735300, 53668332, 147728896, 407729196, 423212608, 516441712, 1227777925, 1323319996, 20440128681, 153088685248, 206158168064, 375868306368, 798666196041

Offset: 1

Author

Paolo P. Lava, Apr 08 2013

Keywords

• nonn
• more

Comments

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.

a(23) > 10^12. - Giovanni Resta, Apr 10 2013

Examples

			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.
		

Crossrefs

Cf. A000203, A001414.

Programs

Maple

with(numtheory); List224346:=proc(q) local a,b,c,j,n;
for n from 1 to q do
a:=ifactors(n)[2]; b:=0; for j from 1 to nops(a) do b:=b+a[j,2]/a[j,1]; od;
c:=sigma(n)/n; if type(b+c,integer) then print(n); fi; od; end:
List224346(10^6);

Mathematica

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

Extensions

a(1)=1 and a(10)-a(22) from Giovanni Resta, Apr 10 2013