login
Numbers k whose average divisor is nonintegral and divides k.
4

%I #23 Jun 08 2020 02:30:39

%S 28,496,8128,950976,2178540,33550336,142990848,301953024,459818240,

%T 675347400,714954240,995248800,1379454720,2701389600,3288789504,

%U 6720569856,8589869056,10200236032,14254365440,30600708096,42763096320,43861478400,66433720320,71271827200

%N Numbers k whose average divisor is nonintegral and divides k.

%C The sequence contains perfect numbers (A000396) and others. Most of them have only small prime factors.

%C The first three terms are in A007691 (multiply perfect numbers) but 950976 is not since sigma_1/k is not an integer.

%C sigma_0(k) is the number of divisors of k (A000005).

%C sigma_1(k) is the sum of the divisors of k [same as sigma(k)] (A000203).

%C Harmonic numbers that are not arithmetic numbers. Of the 937 harmonic numbers below 10^14 there are just 90 such terms, of them 13 are multiply perfect numbers. - _Amiram Eldar_, Jun 08 2020

%H Amiram Eldar, <a href="/A046999/b046999.txt">Table of n, a(n) for n = 1..90</a> (terms below 10^14)

%F Average divisor = m = sigma_1(k)/sigma_0(k) is not an integer but k/m is.

%e k=28, sigma_0=6, sigma_1=56, m=sigma_1/sigma_0=9.333... is not an integer, but k/m=3 is;

%e k=950976, m=2958592/84=3521.333... but k/m=27 is integral.

%Y In A001599 but not in A003601.

%Y Cf. A007691, A046985, A046986, A046987, A000396.

%K nonn

%O 1,1

%A _Labos Elemer_

%E More terms from _Jud McCranie_, Dec 25 2000

%E a(16)-a(24) from _Donovan Johnson_, Apr 22 2008