A001600 Harmonic means of divisors of harmonic numbers.
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
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)
Programs
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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 -
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 *)
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PARI
lista(nn) = for (n=1, nn, if (denominator(q=n*numdiv(n)/sigma(n)) == 1, print1(q, ", "))); \\ Michel Marcus, Jan 13 2016
Extensions
More terms from Matthew Conroy, Jan 15 2006
Comments