A058199 Where d(m) (number of divisors, A000005) falls by at least n.

4, 6, 12, 12, 24, 30, 36, 60, 60, 60, 120, 120, 120, 180, 180, 180, 240, 240, 360, 360, 360, 420, 720, 720, 720, 720, 840, 840, 840, 1260, 1260, 1260, 1680, 1680, 1680, 1680, 1680, 2160, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040

Offset: 1

Author

N. J. A. Sloane, Nov 28 2000

Keywords

• nonn
• nice
• easy

Comments

In the first 500 entries, only 3 entries (1, 2, and 25200) of A002182 are missed. - Bill McEachen, Nov 05 2020

a(n) exists for all n (Turán, 1954). - Amiram Eldar, Apr 13 2024

Examples

			d(12) = 6, d(13) = 2 gives first drop of >= 3, so a(3) = a(4) = 12.
		

References

• József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, p. 39, section II.1.3.a.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2014 (terms 1..500 from T. D. Noe)
• Pál Turán, Problem 71, Matematikai Lapok, Vol. 5 (1954), p. 48, entire volume; Solution to Problem 71, by Lajos Takács, ibid., Vol. 56, (1956), p. 154, entire volume.

Crossrefs

Cf. A000005, A002182, A051950, A058197, A058198.

Programs

Haskell

import Data.List (findIndex)
import Data.Maybe (fromJust)
a058199 n = fromJust $ findIndex (n <=) $ map negate a051950_list
-- Reinhard Zumkeller, Feb 04 2013

Mathematica

max = 10^4; dd = Differences[Table[DivisorSigma[0, m], {m, 1, max}]]; a[n_] := Position[dd, d_ /; d <= -n, 1, 1][[1, 1]]; Table[a[n], {n, 1, -Min[dd] }] (* Jean-François Alcover, Nov 23 2015 *)

Formula

A051950(a(n) + 1) <= - n. - Reinhard Zumkeller, Feb 04 2013

Extensions

More terms from James Sellers, Nov 29 2000