A224912 Numbers m for which 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).

2, 3, 4, 8, 16, 32, 36, 64, 72, 108, 128, 144, 200, 256, 288, 396, 512, 528, 576, 588, 1024, 1040, 1152, 1296, 2000, 2048, 2304, 2320, 2400, 2592, 3888, 4096, 4160, 4608, 4752, 4800, 5184, 5600, 6552, 7200, 8192, 8448, 9216, 9600, 9936, 10368, 11316, 12000

Offset: 1

Author

Paolo P. Lava, Apr 19 2013

Keywords

• nonn

Comments

Apart from 3 all terms are even.

Examples

			Prime factors of 11316 are 2^2, 3, 23 and 41.
Sum_{i=1..5} (p(i)/(p(i)-1)) = 2*(2/(2-1)) + 3/(3-1) + 23/(23-1) + 41/(41-1) = 3331/440.
Sroduct_{i=1..5} (p(i)/(p(i)-1)) = 2*(2/(2-1)) * 3/(3-1) * 23/(23-1) * 41/(41-1) = 2829/440.
Their sum is an integer: 3331/440 + 2829/440 = 14.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A199767, A198391.

Programs

Maple

with(numtheory);
A224912:=proc(i) local b,c,d,n,p;
for n from 2 to i do p:=ifactors(n)[2];
  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);
  if trunc(b)=b then print(n); fi; od; end:
A224912(10^6);