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A230111
Composite numbers m such that 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
8, 10, 64, 512, 720, 800, 1320, 1944, 4096, 5184, 5760, 6400, 7200, 8370, 23520, 32768, 41472, 44000, 46080, 47040, 51200, 69580, 74088, 76096, 84672, 93000, 95040, 105600, 129360, 235200, 240000, 262144, 331776, 368640, 409600, 518400, 546480, 576000, 640000
OFFSET
1,1
EXAMPLE
Prime factors of 7200 are 2^5, 3^2 and 5^2.
Sum_{i=1..9} (p(i)/(p(i)+1)) = 5*(2/(2+1)) + 2*(3/(3+1)) + 2*(5/(5+1)) = 13/2.
Product_{i=1..9} (p(i)/(p(i)-1)) = (2/(2+1))^5 * (3/(3-1))^2 * (5/(5-1))^2 = 225/2.
Their sum is an integer: 13/2 - 225/2 = -106.
MAPLE
with(numtheory); P:=proc(i) local b, 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: P(10^7);
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Oct 09 2013
STATUS
approved