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A109381 Maximum digit of n^2 written in factorial base.

%I A109381 #22 Jan 01 2019 20:16:42
%S A109381 0,1,2,1,2,1,2,2,2,3,4,1,1,2,3,4,2,2,3,3,3,3,4,4,4,5,5,1,2,1,2,2,2,3,
%T A109381 3,4,4,5,2,3,2,2,3,3,4,4,5,3,3,3,4,3,4,5,4,4,4,4,4,5,5,5,5,5,5,5,6,6,
%U A109381 6,6,6,1,1,2,3,4,2,2,3,4,5,3,2,3,4,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,6,2,3,4,2
%N A109381 Maximum digit of n^2 written in factorial base.
%C A109381 Conjecture: lim_{n->infinity} a(n) = infinity. If true, convergence is very slow, since a(1183893) = 1. Sequence is certainly unbounded, since for n >= 4, there is always a square between n*n! and (n+1)!.
%H A109381 Alois P. Heinz, <a href="/A109381/b109381.txt">Table of n, a(n) for n = 0..20954</a>
%e A109381 4^2 = 16 = 2*6+2*2 = 220(factorial base), so a(4) = max(2,2,0) = 2.
%p A109381 b:= proc(n, i) local r; `if`(n<i, n,
%p A109381       max(b(iquo(n, i, 'r'), i+1), r))
%p A109381     end:
%p A109381 a:= n-> b(n^2, 2):
%p A109381 seq(a(n), n=0..188);  # _Alois P. Heinz_, Mar 28 2018
%t A109381 Block[{nn = 105^2, r}, r = Reverse@ Most@ NestWhileList[# + 1 &, 2, #! < nn &]; Array[Max@ IntegerDigits[#^2, MixedRadix@ r] &, Sqrt@ nn, 0]] (* _Michael De Vlieger_, Jan 01 2019 *)
%o A109381 (PARI) f(n, p=2) = if( n<p, n, f( n\p, p+1 )*10 + n%p ); \\ A007623
%o A109381 a(n) = my(dfb=digits(f(n^2))); if (#dfb, vecmax(dfb), 0); \\ _Michel Marcus_, Mar 28 2018
%Y A109381 Cf. A007623, A108731, A301872.
%Y A109381 Indices of 1's in this sequence are A014597.
%K A109381 easy,nonn,base
%O A109381 0,3
%A A109381 _Franklin T. Adams-Watters_, Aug 25 2005