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A001600 Harmonic means of divisors of harmonic numbers.
(Formerly M0609 N0220)
18
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 (list; graph; refs; listen; history; text; internal format)
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.
M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61, (1954). 89-96.
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
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Matthew Conroy, Jan 15 2006
STATUS
approved

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Last modified March 18 22:34 EDT 2024. Contains 370951 sequences. (Running on oeis4.)