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%I #26 Apr 13 2024 05:19:11
%S 1,5,11,11,23,23,47,47,59,59,119,119,167,167,179,179,239,239,359,359,
%T 359,359,719,719,719,719,719,719,839,839,1259,1259,1259,1259,1679,
%U 1679,2519,2519,2519,2519,2519,2519,2519,2519,3359,3359,5039,5039,5039,5039
%N Where d(m) (number of divisors, A000005) rises by at least n.
%C a(n) exists for all n (Turán, 1954). - _Amiram Eldar_, Apr 13 2024
%D 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 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 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 A051950(a(n) + 1) <= n. - _Reinhard Zumkeller_, Feb 04 2013
%e d(11) = 2, d(12) = 6 gives first jump of >= 3, so a(3) = a(4) = 11.
%t 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 With[{d=Differences[DivisorSigma[0,Range[5100]]]},Flatten[Table[ Position[ d,_?(#>=n&),{1},1],{n,50}]]] (* _Harvey P. Dale_, Oct 02 2015 *)
%o (Haskell)
%o import Data.List (findIndex)
%o import Data.Maybe (fromJust)
%o a058197 n = (+ 1) $ fromJust $ findIndex (n <=) $ tail a051950_list
%o -- _Reinhard Zumkeller_, Feb 04 2013
%Y Equals A058198(n) - 1.
%Y Cf. A000005, A051950, A058199.
%K nonn,nice,easy,changed
%O 1,2
%A _N. J. A. Sloane_, Nov 28 2000
%E More terms from _James A. Sellers_, Nov 29 2000
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