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A235224 a(0) = 0, and for n > 0, a(n) = largest k such that A002110(k-1) <= n, where A002110(k) gives the k-th primorial number.

%I A235224 #31 Jul 23 2023 15:43:50
%S A235224 0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,
%T A235224 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A235224 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4
%N A235224 a(0) = 0, and for n > 0, a(n) = largest k such that A002110(k-1) <= n, where A002110(k) gives the k-th primorial number.
%C A235224 For n > 0: a(n) = (length of row n in A235168) = A055642(A049345(n)).
%C A235224 For n > 0, a(n) gives the length of primorial base expansion of n. Also, after zero, each value n occurs A061720(n-1) times. - _Antti Karttunen_, Oct 19 2019
%H A235224 Reinhard Zumkeller, <a href="/A235224/b235224.txt">Table of n, a(n) for n = 0..10000</a>
%H A235224 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A235224 From _Antti Karttunen_, Oct 19 2019: (Start)
%F A235224 a(n) = A061395(A276086(n)).
%F A235224 For all n >= 0, a(n) >= A267263(n).
%F A235224 For all n >= 1, A000040(a(n)) > A328114(n). (End)
%p A235224 A235224 := proc(n)
%p A235224     local k;
%p A235224     if n = 0 then
%p A235224         0;
%p A235224     else
%p A235224         for k from 0 do
%p A235224             if A002110(k-1) > n then
%p A235224                 return k-1 ;
%p A235224             end if;
%p A235224         end do:
%p A235224     end if;
%p A235224 end proc: # _R. J. Mathar_, Apr 19 2021
%t A235224 primorial[n_] := Times @@ Prime[Range[n]];
%t A235224 a[n_] := TakeWhile[primorial /@ Range[0, n], # <= n &] // Length;
%t A235224 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Oct 27 2021 *)
%o A235224 (Haskell)
%o A235224 a235224 n = length $ takeWhile (<= n) a002110_list
%o A235224 (PARI) A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); }; \\ _Antti Karttunen_, Oct 19 2019
%o A235224 (PARI) A235224(n, p=2) = if(!n,n,if(n<p, 1, 1+A235224(n\p, nextprime(p+1)))); \\ (Recursive implementation) - _Antti Karttunen_, Oct 19 2019
%Y A235224 Cf. A000040, A002110, A049345, A055642, A061395, A061720, A084558, A267263, A276086, A235168, A328114, A328404, A328405, A328406.
%K A235224 nonn
%O A235224 0,3
%A A235224 _Reinhard Zumkeller_, Jan 05 2014
%E A235224 Name corrected to match the data by _Antti Karttunen_, Oct 19 2019