A058197 Where d(m) (number of divisors, A000005) rises by at least n.

1, 5, 11, 11, 23, 23, 47, 47, 59, 59, 119, 119, 167, 167, 179, 179, 239, 239, 359, 359, 359, 359, 719, 719, 719, 719, 719, 719, 839, 839, 1259, 1259, 1259, 1259, 1679, 1679, 2519, 2519, 2519, 2519, 2519, 2519, 2519, 2519, 3359, 3359, 5039, 5039, 5039, 5039

Offset: 1

Author

N. J. A. Sloane, Nov 28 2000

Keywords

• nonn
• nice
• easy

Comments

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

Examples

			d(11) = 2, d(12) = 6 gives first jump of >= 3, so a(3) = a(4) = 11.
		

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..2044 (terms 1..1004 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

Equals A058198(n) - 1.

Cf. A000005, A051950, A058199.

Programs

Haskell

import Data.List (findIndex)
import Data.Maybe (fromJust)
a058197 n = (+ 1) $ fromJust $ findIndex (n <=) $ tail a051950_list
-- Reinhard Zumkeller, Feb 04 2013

Mathematica

d[m_] := d[m] = DivisorSigma[0, m]; td = Table[d[m] - d[m-1], {m, 2, 6000}]; a[n_] := Position[td, j_ /; j >= n, 1][[1, 1]]; Table[a[n], {n, Max[td]}] (* Jean-François Alcover, Nov 02 2011 *)
With[{d=Differences[DivisorSigma[0,Range[5100]]]},Flatten[Table[ Position[ d,?(#>=n&),{1},1],{n,50}]]] (* _Harvey P. Dale, Oct 02 2015 *)

Formula

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

Extensions

More terms from James Sellers, Nov 29 2000