A334168 Numbers m whose leading digit and the other decimal digits appear respectively before and directly after the decimal point of its m-th root.

1, 12, 104, 1006, 10009, 100011, 1000013, 10000016, 100000018, 1000000020, 10000000023, 100000000025, 1000000000027, 10000000000029, 100000000000032, 1000000000000034, 10000000000000036, 100000000000000039, 1000000000000000041, 10000000000000000043

Offset: 1

Author

Alaa A. Nasr, Apr 17 2020

Keywords

• nonn
• base

Comments

These numbers m are equal to the integer part of their m-th root multiplied by 10 to the power of the number of digits of m-1.

The last nonzero digits of the terms are conjectured to be the Beatty sequence for log(10), A059545.

Examples

			12^(1/12) = 1.23007..., 104^(1/104) = 1.0457..., 100011^(1/100011) = 1.00011512...
		

Crossrefs

Cf. A059545.

Programs

PARI

isok(m) = floor(10^(#Str(m)-1)*sqrtn(m, m)) == m; \\ Michel Marcus, Apr 17 2020

Python

# for the first 6 terms
import math
for n in range(1, 1000000):
    if math.floor((n**(1/n))*10**(len(str(n))-1)) == n:
        print(n)

Formula

a(n+1) = floor((10^(n/10^n))*10^n) (conjectured).

a(n) = 10^(n-1) + floor((n-1)*log(10)) (conjectured). - David A. Corneth, Apr 19 2020

Extensions

More terms from David A. Corneth, Apr 19 2020