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Numbers m whose leading digit and the other decimal digits appear respectively before and directly after the decimal point of its m-th root.
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%I #38 Mar 30 2023 07:29:27

%S 1,12,104,1006,10009,100011,1000013,10000016,100000018,1000000020,

%T 10000000023,100000000025,1000000000027,10000000000029,

%U 100000000000032,1000000000000034,10000000000000036,100000000000000039,1000000000000000041,10000000000000000043

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

%C 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.

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

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

%F a(n) = 10^(n-1) + floor((n-1)*log(10)) (conjectured). - _David A. Corneth_, Apr 19 2020

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

%o (Python)

%o # for the first 6 terms

%o import math

%o for n in range(1, 1000000):

%o if math.floor((n**(1/n))*10**(len(str(n))-1)) == n:

%o print(n)

%o (PARI) isok(m) = floor(10^(#Str(m)-1)*sqrtn(m, m)) == m; \\ _Michel Marcus_, Apr 17 2020

%Y Cf. A059545.

%K nonn,base

%O 1,2

%A _Alaa A. Nasr_, Apr 17 2020

%E More terms from _David A. Corneth_, Apr 19 2020