A090240 - OEIS

%I #15 Apr 15 2024 03:31:16

%S 1,2,3,5,6,7,8,9,10,11,13,14,15,17,19,21,24,25,26,27,29,31,35,37,39,

%T 41,42,44,45,46,47,48,49,50,51,53,54,60,61,70,73,75,77,78,80,81,82,83,

%U 84,85,86,87,88,89,91,92,94,96,97,99,101,102,105,106,107,108,110,114,115

%N Numbers which occur as the harmonic mean of the divisors of m for some m.

%C The equation n = m*tau(m)/sigma(m) has an integer solution m.

%C Here tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).

%C A001600 sorted in order.

%C The Mersenne exponents (A000043) are in this sequence because the even perfect numbers, 2^(p-1)*(2^p-1) where p is in A000043, are all harmonic numbers (A001599) with harmonic mean of divisors p. - _Amiram Eldar_, Apr 15 2024

%D For further references see A001599.

%H T. Goto and S. Shibata, <a href="http://dx.doi.org/10.1090/S0025-5718-03-01554-0">All numbers whose positive divisors have integral harmonic mean up to 300</a>, Math. Comput. 73 (2004), 475-491.

%t f[n_] := (n*DivisorSigma[0, n]/DivisorSigma[1, n]); a = Table[ 0, {120}]; Do[ b = f[n]; If[ IntegerQ[b] && b < 121 && a[[b]] == 0, a[[b]] = n], {n, 1, 560000000}]; Select[ Range[120], a[[ # ]] > 0 &] (* _Robert G. Wilson v_, Feb 14 2004 *)

%Y Values of m are in A091911.

%Y Complement of A157849.

%Y Cf. A000043, A001599, A001600, A090758, A090759, A090760, A090761, A090762.

%K nonn

%O 1,2

%A _R. K. Guy_, Feb 08 2004

%E More terms from _Robert G. Wilson v_, Feb 14 2004