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

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, 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, 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

Offset: 0

Author

Franklin T. Adams-Watters, Aug 25 2005

Keywords

• easy
• nonn
• base

Comments

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)!.

Examples

			4^2 = 16 = 2*6+2*2 = 220(factorial base), so a(4) = max(2,2,0) = 2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20954

Crossrefs

Cf. A007623, A108731, A301872.

Indices of 1's in this sequence are A014597.

Programs

Maple

b:= proc(n, i) local r; `if`(n b(n^2, 2):
seq(a(n), n=0..188);  # Alois P. Heinz, Mar 28 2018

Mathematica

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 *)

PARI

f(n, p=2) = if( nA007623
a(n) = my(dfb=digits(f(n^2))); if (#dfb, vecmax(dfb), 0); \\ Michel Marcus, Mar 28 2018