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

1,1

Comments

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

Links

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

Example

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.

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

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

Sequence in context: A007939 A212767 A126807 * A223587 A302312 A245199

Adjacent sequences: A230107 A230108 A230109 * A230111 A230112 A230113

Keyword

nonn

Author

Paolo P. Lava, Oct 09 2013

Status

approved