A097326 Largest integer m such that m*n has the same decimal digit length as n.

9, 4, 3, 2, 1, 1, 1, 1, 1, 9, 9, 8, 7, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9

Offset: 1

Author

Rick L. Shepherd, Aug 04 2004

Keywords

• base
• easy
• nonn

Comments

For any positive base B >= 2 the corresponding sequence contains only terms from 1 to B-1 inclusive so the corresponding sequence for binary is all 1's (A000012).

Examples

			a(12)=8 as 12 and 8*12=96 both have two decimal digits while 9*12=108 has three.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A061601 (analog for decimal m+n), A035327 (analog for binary m+n), A097327.

Cf. A055642.

Programs

Mathematica

limn[n_]:=Module[{k=9,len=IntegerLength[n]},While[IntegerLength[k*n] > len, k--];k]; Array[limn,110] (* Harvey P. Dale, Apr 28 2018 *)
Table[Ceiling[10^IntegerLength[n]/n] - 1, {n, 100}] (* Paolo Xausa, Nov 06 2024 *)

PARI

a(n) = my(m=1, sn=#Str(n)); while (#Str(m*n) <= sn, m++); m-1; \\ Michel Marcus, Oct 05 2021

Python

def a(n): return (10**len(str(n))-1)//n
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Oct 05 2021

Formula

a(n) = A097327(n) - 1.

a(n) = floor((10^A055642(n) - 1)/n). - Michael S. Branicky, Oct 05 2021