login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Numbers m such that there is at least one smaller number k < m with the same harmonic mean of divisors as m.
2

%I #17 Oct 30 2021 10:42:25

%S 135,224,496,936,1485,1488,1755,2295,2464,2565,2912,3105,3808,3915,

%T 4185,4256,4464,4680,4995,5152,5456,5535,5805,6345,6448,6496,6552,

%U 6860,6944,7155,7965,8235,8288,8432,9045,9184,9424,9585,9632,9855,10296,10528,10665,10976

%N Numbers m such that there is at least one smaller number k < m with the same harmonic mean of divisors as m.

%C The corresponding values of k (the least in case there are more than one) are 84, 120, 140, 864, 924, 420, 1092, 1428, 1320, 1596, ... (see the link for more values).

%C The least term m with more than one smaller number k with the same harmonic mean of divisors as m is m = a(1237) = A348697(1) = 321048 with k = 201096 and 296352.

%H Amiram Eldar, <a href="/A348696/b348696.txt">Table of n, a(n) for n = 1..10000</a>

%H Amiram Eldar, <a href="/A348696/a348696.txt">Table of n, a(n), least k for n=1..1000</a>

%e 135 is a term since the harmonic mean of divisors of 135 is 9/2, and it is also the harmonic mean of divisors of 84 which is smaller than 135.

%t h = Table[DivisorSigma[0, n]/DivisorSigma[-1, n], {n, 1, 10000}]; i = Position[(t = Tally[h])[[;; , 2]], _?(# > 1 &)] // Flatten; Position[h, #][[2 ;; -1]] & /@ t[[i, 1]] // Flatten // Sort

%Y Cf. A099377, A099378, A348697.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 30 2021