A226476 Numbers n with the property that, if tau(n) = k = number of divisors of n, and the d(i) are the divisors [arranged in increasing order], then the sum 1/d(k) + 1/d(k-1) + 1/d(k-2) + ... + 1/d(q) is an integer for some q.

1, 6, 24, 28, 120, 496, 672, 2016, 4320, 4680, 8128, 8190, 26208, 30240, 32760, 42336, 45864, 392448, 523776, 714240, 1571328, 2178540, 8910720, 17428320, 20427264, 23569920, 29795040, 33550336, 34369920, 45532800, 61900800

Offset: 1

Author

Michel Lagneau, Jun 11 2013

Keywords

• nonn

Comments

By convention, for n = 1, a(1) = 1 with q = 1.

The corresponding pairs (tau(n), q) are (1, 1), (4, 2), (8, 3), (6, 2), (16, 2), (10, 2), (24, 2), (36, 6), (48, 3), (48, 3), (14, 2), (48, 6), (72, 3), (96, 2), (96, 2), (72, 7), (72, 7), (72, 5), (80, 2), (120, 8), (120, 6), (216, 2), (384, 3), (432, 3), (240, 3), (320, 2), (360, 5), (26, 2), (384, 5), (384, 2), (288, 9).

Properties of this sequence:

q = 2 if n = 1, 6, 28, 120, 496, 672, 8128, ... is a multiply-perfect number (see A007691 where it is conjectured that this sequence is infinite), which would imply that this sequence is also infinite because A007691 is a subsequence.

Examples

			24 is in the sequence because the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24, and the sum 1/24 + 1/12 + 1/8 + 1/6 + 1/4 + 1/3 = 1.
28 is in the sequence because 28 is a multiply-perfect number: the divisors are 1, 2, 4, 7, 14, 28, and the sum of the reciprocals of all the divisors is 1/28 + 1/14 + 1/7 + 1/4 + 1/2 + 1 = 2.
		

Crossrefs

Cf. A000005, A000203, A007691, A225110.

Programs

Maple

with(numtheory): for n from 1 to 10000000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from n1 by -1 to 1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) then ii:=1: printf(`%d, `,n):else fi:od:od:

Extensions

Edited by Jon E. Schoenfield and N. J. A. Sloane, Sep 09 2017