A001600 - OEIS

%I M0609 N0220 #47 Dec 19 2021 10:02:25

%S 1,2,3,5,6,5,8,9,11,10,7,15,15,14,17,24,24,21,13,19,27,25,29,26,44,44,

%T 29,46,39,46,27,42,47,47,54,35,41,60,51,37,48,45,49,50,49,53,77,86,86,

%U 51,96,75,70,80,99,110,81,84,13,102,82,96,114,53,108,115,105,116,91,85,105

%N Harmonic means of divisors of harmonic numbers.

%C Values of n*tau(n)/sigma(n) corresponding to terms of A001599, where tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).

%C Kanold (1957) proved that each term appears only a finite number of times. - _Amiram Eldar_, Jun 01 2020

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H R. J. Mathar, <a href="/A001600/b001600.txt">Table of n, a(n) for n = 1..937</a>, extending the former b-file of T. D. Noe.

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, <a href="http://arxiv.org/abs/1601.03081">The Biharmonic mean</a>, arXiv:1601.03081 [math.NT], 2016.

%H G. L. Cohen, <a href="/A007340/a007340.pdf">Email to N. J. A. Sloane, Apr. 1994</a>

%H M. Garcia, <a href="http://www.jstor.org/stable/2307792">On numbers with integral harmonic mean</a>, Amer. Math. Monthly 61, (1954). 89-96.

%H Takeshi Goto, <a href="http://www.ma.noda.tus.ac.jp/u/tg/html/harmonic-e.html#mark1">All harmonic numbers less than 10^14</a>

%H Takeshi Goto, <a href="http://www.ma.noda.tus.ac.jp/u/tg/files/list4">Table of a(n) for n=1..937</a>

%H Hans-Joachim Kanold , <a href="http://dx.doi.org/10.1007/BF01342887">Über das harmonische Mittel der Teiler einer natürlichen Zahl</a>, Math. Ann., Vol. 133 (1957), pp. 371-374.

%H O. Ore, <a href="http://www.jstor.org/stable/2305616">On the averages of the divisors of a number</a>, Amer. Math. Monthly, 55 (1948), 615-619.

%H O. Ore, <a href="/A001599/a001599.pdf">On the averages of the divisors of a number</a> (annotated scanned copy)

%t A001600 = Reap[Do[tau = DivisorSigma[0, n]; sigma = DivisorSigma[1, n]; h = n*tau/sigma; If[IntegerQ[h], Print[h]; Sow[h]], {n, 1, 90000000}]][[2, 1]](* _Jean-François Alcover_, May 11 2012 *)

%o (Haskell)

%o a001600 n = a001600_list !! (n-1)

%o a001600_list =

%o    [numerator m | x <- [1..], let m = hm x, denominator m == 1] where

%o    hm x = genericLength divs * recip (sum $ map recip divs)

%o           where divs = map fromIntegral $ a027750_row x

%o -- _Reinhard Zumkeller_, Apr 01 2014

%o (PARI) lista(nn) = for (n=1, nn, if (denominator(q=n*numdiv(n)/sigma(n)) == 1, print1(q, ", "))); \\ _Michel Marcus_, Jan 13 2016

%Y Cf. A001599, A090240 (sorted values).

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Matthew Conroy_, Jan 15 2006