A058198 -id:A058198 - cp's OEIS frontend

A051950 Differences between values of tau(n) (A000005): a(n) = tau(n)-tau(n-1).

1, 0, 1, -1, 2, -2, 2, -1, 1, -2, 4, -4, 2, 0, 1, -3, 4, -4, 4, -2, 0, -2, 6, -5, 1, 0, 2, -4, 6, -6, 4, -2, 0, 0, 5, -7, 2, 0, 4, -6, 6, -6, 4, 0, -2, -2, 8, -7, 3, -2, 2, -4, 6, -4, 4, -4, 0, -2, 10, -10, 2, 2, 1, -3, 4, -6, 4, -2, 4, -6, 10, -10, 2, 2, 0, -2, 4, -6, 8, -5, -1, -2, 10, -8, 0, 0, 4, -6, 10

Offset: 2

Asher Auel, Dec 16 1999

N. J. A. Sloane, Table of n, a(n) for n = 2..20000 [First 10000 terms from T. D. Noe]

Cf. A000005, A058197, A058198, A058199.

a051950 n = a051950_list !! n
a051950_list = 1 : zipWith (-) (tail a000005_list) a000005_list
-- Reinhard Zumkeller, Feb 04 2013

Differences[DivisorSigma[0, Range[100]]] (* Alonso del Arte, Oct 14 2013, based on T. D. Noe's program for A182394 *)

G.f.: (1-x)*sum(k >= 1, x^k/(1-x^k)). - Benoit Cloitre, Apr 24 2003

Edited (and offset corrected) by N. J. A. Sloane, Oct 05 2017

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

Original entry on oeis.org

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

N. J. A. Sloane, Nov 28 2000

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

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

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.

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.

Equals A058198(n) - 1.

Cf. A000005, A051950, A058199.

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

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 *)

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

More terms from James Sellers, Nov 29 2000

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

Original entry on oeis.org

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

N. J. A. Sloane, Nov 28 2000

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

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

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.

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.

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

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

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 *)

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

More terms from James Sellers, Nov 29 2000

cp's OEIS Frontend

A051950 Differences between values of tau(n) (A000005): a(n) = tau(n)-tau(n-1).

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

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

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Haskell

Mathematica

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Extensions