cp's OEIS Frontend

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.

%I A226476 #29 Sep 16 2017 00:28:55
%S A226476 1,6,24,28,120,496,672,2016,4320,4680,8128,8190,26208,30240,32760,
%T A226476 42336,45864,392448,523776,714240,1571328,2178540,8910720,17428320,
%U A226476 20427264,23569920,29795040,33550336,34369920,45532800,61900800
%N 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.
%C A226476 By convention, for n = 1, a(1) = 1 with q = 1.
%C A226476 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).
%C A226476 Properties of this sequence:
%C A226476 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.
%e A226476 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.
%e A226476 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.
%p A226476 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:
%Y A226476 Cf. A000005, A000203, A007691, A225110.
%K A226476 nonn
%O A226476 1,2
%A A226476 _Michel Lagneau_, Jun 11 2013
%E A226476 Edited by _Jon E. Schoenfield_ and _N. J. A. Sloane_, Sep 09 2017