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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..31.

EXAMPLE

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.

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:

CROSSREFS

Cf. A000005, A000203, A007691, A225110.

Sequence in context: A118372 A263928 A219362 * A216793 A294900 A064510

Adjacent sequences:  A226473 A226474 A226475 * A226477 A226478 A226479

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jun 11 2013

EXTENSIONS

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

STATUS

approved

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Last modified July 22 07:23 EDT 2019. Contains 325216 sequences. (Running on oeis4.)