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A307600 Numbers k such that the digits of k^(1/4) begin with k. 9
0, 1, 21, 463, 464, 9999, 10000, 215443, 4641588, 99999999, 100000000, 2154434689, 2154434690, 46415888335, 46415888336, 999999999999, 1000000000000, 21544346900318, 464158883361277, 9999999999999999, 10000000000000000, 215443469003188371, 215443469003188372 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Program is in A307371.

From Bernard Schott, May 01 2019: (Start)

There are two nontrivial families in this sequence:

1st: 21, 215443, 2154434689, 2154434690, 21544346900318, ...

2nd: 463, 464, 4641588, 46415888335, 46415888336, ... (End)

From Jon E. Schoenfield, May 04 2019: (Start)

For each number k such that the digits of k^(1/m) begin with k, we have, for each m >= 2, floor(k^(1/m) * 10^d) = k for some integer d, so k^(1/m) * 10^d ~= k; solving for k gives k ~= 10^(d*m/(m-1)).

In the m=4 case (this sequence), this gives k ~= 10^(d*4/3) so, as d is incremented by 1, 10^(d*4/3) increases by a factor of 10^(4/3) = 10000^1/3 = 21.5443469...:

.

   d |     10^(d*4/3)

  ---+---------------------

   0 |             1

   1 |            21.544...

   2 |           464.158...

   3 |         10000

   4 |        215443.469...

   5 |       4641588.833...

   6 |     100000000

   7 |    2154434690.031...

   8 |   46415888336.127...

   9 | 1000000000000

.

Each nonnegative integer d corresponds to one or two terms in the sequence. Letting j = floor(10000^(d/3)), j is necessarily a term; j-1 is also a term iff (j-1)^(1/4)*10^d < j. This inequality is satisified

   for d == 1 (mod 3) at d = 7, 13, 16, 34, 37, ...;

   for d == 2 (mod 3) at d = 2, 8, 20, 29, 32, 35, ...;

   and at every d == 0 (mod 3).

(The sequence contains no other terms than numbers k of the form j or j-1 where j = floor(10000^(d/3)) for some nonnegative integer d.)

(End)

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1172

EXAMPLE

215443^(1/4) = 21.544335..., which begins with "215443", so 215443 is in the sequence.

CROSSREFS

Cf. A307371, A307588.

Cf. A052211 (analog for 4th power instead of 1/4).

Sequence in context: A076552 A126996 A158603 * A025603 A296586 A269922

Adjacent sequences:  A307597 A307598 A307599 * A307601 A307602 A307603

KEYWORD

nonn,base

AUTHOR

Dmitry Kamenetsky, Apr 17 2019

EXTENSIONS

a(12)-a(23) from Jon E. Schoenfield, May 01 2019

STATUS

approved

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Last modified April 9 20:55 EDT 2020. Contains 333363 sequences. (Running on oeis4.)