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A116998 Numbers having no fewer distinct prime factors than any predecessor; a(1) = 1.

%I A116998 #47 Nov 14 2024 14:05:17
%S A116998 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 A116998 102,105,110,114,120,126,130,132,138,140,150,154,156,165,168,170,174,
%U A116998 180,182,186,190,195,198,204,210,330,390,420,462,510,546,570,630,660
%N A116998 Numbers having no fewer distinct prime factors than any predecessor; a(1) = 1.
%C A116998 A001221(a(n)) <= A001221(a(n+1));
%C A116998 A002110 is a subsequence.
%C A116998 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 A116998 Called omega-largely composite numbers by Erdős and Nicolas (1981). - _Amiram Eldar_, Jun 24 2023
%H A116998 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 A116998 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 A116998 a:= proc(n) option remember; local k, t;
%p A116998       t:= nops(ifactors(a(n-1))[2]);
%p A116998       for k from 1+a(n-1) while nops(ifactors(k)[2])<t do od; k
%p A116998     end: a(1):=1:
%p A116998 seq(a(n), n=1..80);  # _Alois P. Heinz_, Oct 05 2012
%t A116998 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 A116998 Cf. A029744, A067128.
%Y A116998 Cf. A001221 (omega), A002110 (primorial numbers).
%K A116998 nonn,look
%O A116998 1,2
%A A116998 _Reinhard Zumkeller_, Apr 03 2006