A001600 Harmonic means of divisors of harmonic numbers. (Formerly M0609 N0220)

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, 29, 46, 39, 46, 27, 42, 47, 47, 54, 35, 41, 60, 51, 37, 48, 45, 49, 50, 49, 53, 77, 86, 86, 51, 96, 75, 70, 80, 99, 110, 81, 84, 13, 102, 82, 96, 114, 53, 108, 115, 105, 116, 91, 85, 105

Offset

1,2

Comments

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).

Kanold (1957) proved that each term appears only a finite number of times. - Amiram Eldar, Jun 01 2020

References

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

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

Links

R. J. Mathar, Table of n, a(n) for n = 1..937, extending the former b-file of T. D. Noe.

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.

G. L. Cohen, Email to N. J. A. Sloane, Apr. 1994

M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61, (1954). 89-96.

Takeshi Goto, All harmonic numbers less than 10^14

Takeshi Goto, Table of a(n) for n=1..937

Hans-Joachim Kanold , Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371-374.

O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.

O. Ore, On the averages of the divisors of a number (annotated scanned copy)

Mathematica

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 *)

Prog

(Haskell)
a001600 n = a001600_list !! (n-1)
a001600_list =
   [numerator m | x <- [1..], let m = hm x, denominator m == 1] where
   hm x = genericLength divs * recip (sum $ map recip divs)
          where divs = map fromIntegral $ a027750_row x
-- Reinhard Zumkeller, Apr 01 2014
(PARI) lista(nn) = for (n=1, nn, if (denominator(q=n*numdiv(n)/sigma(n)) == 1, print1(q, ", "))); \\ Michel Marcus, Jan 13 2016

Crossrefs

Cf. A001599, A090240 (sorted values).

Sequence in context: A306233 A254105 A247548 * A175578 A347861 A316609

Adjacent sequences: A001597 A001598 A001599 * A001601 A001602 A001603

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Matthew Conroy, Jan 15 2006

Status

approved