A230110 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).

8, 10, 30, 63, 64, 512, 588, 720, 800, 1320, 3960, 4096, 5184, 5760, 6400, 7200, 21600, 27720, 27900, 32768, 35280, 41472, 46080, 51200, 70840, 84672, 92400, 95040, 105600, 151200, 188160, 262144, 331776, 368640, 376320, 409600, 504000, 518400, 576000, 640000

Offset: 1

Author

Paolo P. Lava, Oct 09 2013

Keywords

• nonn

Comments

Includes 2^(3*a) * 3^(4*b) if 3*a >= 4*b. - Robert Israel, Mar 30 2023

Examples

			Prime factors of 3960 are 2^3, 3^2, 5 and 11.
Sum_{i=1..7} (p(i)/(p(i)+1)) = 3*(2/(2+1)) + 2*(3/(3+1)) + 5/(5+1) + 11/(11+1) = 21/4.
Product_{i=1..7} (p(i)/(p(i)-1)) = (2/(2+1))^3 * (3/(3-1))^2 * 5/(5-1) * 11/(11-1) = 99/4.
Their sum is an integer: 21/4 + 99/4 = 30.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..155

Crossrefs

Cf. A199767, A198391, A227034, A227248, A230111, A230112.

Programs

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);