A230112 Composite numbers m such that Product_{i=1..k} (p_i/(p_i-1)) / Sum_{i=1..k} (p_i/(p_i+1)) is an integer, where p_i are the k prime factors of m (with multiplicity).

4, 8, 16, 64, 256, 720, 800, 2200, 4096, 25600, 33600, 36288, 41472, 46080, 65536, 92400, 104960, 235200, 282240, 338688, 376320, 403200, 419840, 535680, 556640, 576000, 580800, 640000, 844800, 979776, 1088640, 1244160, 1354752, 1382400, 1505280, 1689600, 1995840

Offset: 1

Author

Paolo P. Lava, Oct 09 2013

Keywords

• nonn

Examples

			Prime factors of 2200 are 2^3, 5^2 and 11.
Sum_{i=1..6} (p(i)/(p(i)+1)) = 3*(2/(2+1)) + 2*(5/(5+1)) + 11/(11+1) = 55/12.
Product_{i=1..6} (p(i)/(p(i)-1)) = (2/(2-1))^3*(5/(5-1))^2*11/(11-1) = 55/4.
The ratio is an integer: (55/4) / (55/12) = 3.
		

Crossrefs

Cf. A199767, A198391, A227034, A227248, A230110, A230111.

Programs

Maple

with(numtheory); P:=proc(q) local a, d, n, p;
for n from 2 to q do if not isprime(n) then p:=ifactors(n)[2];
a:=mul((op(1, d)/(op(1, d)-1))^op(2, d), d=p)/add((op(1, d)/(op(1, d)+1))*op(2, d), d=p); if type(a, integer) then print(n); fi; fi;
od; end: P(10^7);