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

0,3

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)!.

Links

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

Example

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

Maple

b:= proc(n, i) local r; `if`(n<i, n,
      max(b(iquo(n, i, 'r'), i+1), r))
    end:
a:= 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 *)

Prog

(PARI) f(n, p=2) = if( n<p, n, f( n\p, p+1 )*10 + n%p ); \\ A007623
a(n) = my(dfb=digits(f(n^2))); if (#dfb, vecmax(dfb), 0); \\ Michel Marcus, Mar 28 2018

Crossrefs

Cf. A007623, A108731, A301872.

Indices of 1's in this sequence are A014597.

Sequence in context: A133989 A029398 A025817 * A058506 A325509 A324117

Adjacent sequences: A109378 A109379 A109380 * A109382 A109383 A109384

Keyword

easy,nonn,base

Author

Franklin T. Adams-Watters, Aug 25 2005

Status

approved