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

%I A058197 #27 Jul 02 2025 16:02:00
%S A058197 1,5,11,11,23,23,47,47,59,59,119,119,167,167,179,179,239,239,359,359,
%T A058197 359,359,719,719,719,719,719,719,839,839,1259,1259,1259,1259,1679,
%U A058197 1679,2519,2519,2519,2519,2519,2519,2519,2519,3359,3359,5039,5039,5039,5039
%N A058197 Where d(m) (number of divisors, A000005) rises by at least n.
%C A058197 a(n) exists for all n (Turán, 1954). - _Amiram Eldar_, Apr 13 2024
%D A058197 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.
%H A058197 Amiram Eldar, <a href="/A058197/b058197.txt">Table of n, a(n) for n = 1..2044</a> (terms 1..1004 from T. D. Noe)
%H A058197 Pál Turán, Problem 71, Matematikai Lapok, Vol. 5 (1954), p. 48, <a href="https://real-j.mtak.hu/9380">entire volume</a>; Solution to Problem 71, by Lajos Takács, ibid., Vol. 56, (1956), p. 154, <a href="https://real-j.mtak.hu/9386">entire volume</a>.
%F A058197 A051950(a(n) + 1) <= n. - _Reinhard Zumkeller_, Feb 04 2013
%e A058197 d(11) = 2, d(12) = 6 gives first jump of >= 3, so a(3) = a(4) = 11.
%t A058197 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 *)
%t A058197 With[{d=Differences[DivisorSigma[0,Range[5100]]]},Flatten[Table[ Position[ d,_?(#>=n&),{1},1],{n,50}]]] (* _Harvey P. Dale_, Oct 02 2015 *)
%o A058197 (Haskell)
%o A058197 import Data.List (findIndex)
%o A058197 import Data.Maybe (fromJust)
%o A058197 a058197 n = (+ 1) $ fromJust $ findIndex (n <=) $ tail a051950_list
%o A058197 -- _Reinhard Zumkeller_, Feb 04 2013
%Y A058197 Equals A058198(n) - 1.
%Y A058197 Cf. A000005, A051950, A058199.
%K A058197 nonn,nice,easy
%O A058197 1,2
%A A058197 _N. J. A. Sloane_, Nov 28 2000
%E A058197 More terms from _James Sellers_, Nov 29 2000