A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

0, 1, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 192, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 1728, 1920, 2160, 2304, 2880, 3456, 4320, 5760, 6912, 8640, 11520, 17280, 23040, 25920, 30240, 34560, 46080, 51840, 60480, 69120, 86400, 103680, 120960

Offset

1,3

Comments

The corresponding record values are 0, 1, 2, 3, 4, 8, 20, 26, 48, 76, 112, 132, 208, ... (see the link for more values).

Deléglise et al. (2008) proved that the number of powerful (A001694) terms in this sequence is finite. They ask whether a(391) = 485432135516160000 (the 112th powerful term) is the largest. - Amiram Eldar, Aug 20 2019

Is abs(omega(a(n)) - omega(a(n+1))) <= 1? (Cf. A001221.) - David A. Corneth, Apr 16 2020

Links

David A. Corneth, Table of n, a(n) for n = 1..884 (terms 1..762 from Amiram Eldar, calculated by Deléglise et al.)

David A. Corneth, Table of n, a(n) for n = 1..1544 if abs(omega(a(n)) - omega(a(n + 1))) <= 1

M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár, Journal of Number Theory, Vol. 128, No. 6 (2008), pp. 1676-1716.

M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Tables des 761 premiers champions de la fonction de Kalmar, alternative link.

Amiram Eldar, Table of n, a(n), A074206(a(n)), calculated by Deléglise et al.

T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Formula

For n >= 1, a(1+n) = A108951(A330686(n)). - Antti Karttunen, Dec 31 2019

Mathematica

a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; s = {}; am=-1; Do[a1 = a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 0, 10000}]; s

Crossrefs

Cf. A001221, A001694, A002093, A033833, A074206, A163272, A330686 (after primorial deflation).

Sequence in context: A176777 A318761 A020153 * A338927 A286373 A286372

Adjacent sequences: A307863 A307864 A307865 * A307867 A307868 A307869

Keyword

nonn

Author

Amiram Eldar, May 02 2019

Status

approved