A094229 Numbers n such that d(n) >= n-th harmonic number H(n)=sum_{i=1..n}1/i.

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 128, 130, 132

Offset

1,2

Comments

A positive integer n belongs to the sequence if and only the number of its divisors (d(n)) is >= the average number of divisors, in the range from 1 through n, for all positive integers (H(n)).

Visible sharp bend on the graph around the 800th term occur where the n-th harmonic number exceeds 8. - Ivan Neretin, Oct 16 2016

Links

Ivan Neretin, Table of n, a(n) for n = 1..5000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 840. (d(n) is given as sigma_0(n).)

Example

6 is in the sequence because the number of its divisors, 4, is greater than the 6th harmonic number, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 2.45.

Mathematica

ok[n_] := DivisorSigma[0, n] >= HarmonicNumber[n]; Select[ Range[132], ok] (* Jean-François Alcover, Sep 19 2011 *)

Crossrefs

d(n)=A000005(n), H(n)=A001008(n)/A002805(n). See also A004080.

Sequence in context: A063450 A353684 A047229 * A067290 A106577 A068577

Adjacent sequences: A094226 A094227 A094228 * A094230 A094231 A094232

Keyword

nonn

Author

Matthew Vandermast, May 29 2004

Status

approved