A116998 - OEIS

%I #47 Nov 14 2024 14:05:17

%S 1,2,3,4,5,6,10,12,14,15,18,20,21,22,24,26,28,30,42,60,66,70,78,84,90,

%T 102,105,110,114,120,126,130,132,138,140,150,154,156,165,168,170,174,

%U 180,182,186,190,195,198,204,210,330,390,420,462,510,546,570,630,660

%N Numbers having no fewer distinct prime factors than any predecessor; a(1) = 1.

%C A001221(a(n)) <= A001221(a(n+1));

%C A002110 is a subsequence.

%C The unitary version of Ramanujan's largely composite numbers (A067128), numbers having no fewer unitary divisors than any predecessor. - _Amiram Eldar_, Jun 08 2019

%C Called omega-largely composite numbers by Erdős and Nicolas (1981). - _Amiram Eldar_, Jun 24 2023

%H Amiram Eldar, <a href="/A116998/b116998.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2500 from Alois P. Heinz)

%H Paul Erdős and Jean-Louis Nicolas, <a href="https://static.renyi.hu/~p_erdos/1981-34.pdf">Sur la fonction: nombre de facteurs premiers de n</a>, L'Enseignement Math., Vol. 27 (1981), pp. 3-27; <a href="http://math.univ-lyon1.fr/~nicolas/ensmathErdos81.pdf">alternative link</a>.

%p a:= proc(n) option remember; local k, t;

%p       t:= nops(ifactors(a(n-1))[2]);

%p       for k from 1+a(n-1) while nops(ifactors(k)[2])<t do od; k

%p     end: a(1):=1:

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Oct 05 2012

%t a[1] = 1; a[n_] := a[n] = For[nu = PrimeNu[a[n-1]]; k = a[n-1]+1, True, k++, If[PrimeNu[k] >= nu, Return[k]]]; Array[a, 80] (* _Jean-François Alcover_, Apr 11 2017 *)

%Y Cf. A029744, A067128.

%Y Cf. A001221 (omega), A002110 (primorial numbers).

%K nonn,look

%O 1,2

%A _Reinhard Zumkeller_, Apr 03 2006