Whale number

In entertaining mathematics, the Kit number is a number from :

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, ... (sequence A007629 in OEIS )

Keith numbers were introduced by in 1987 ^[1] . Numbers are hard to find; for 2017, only 100 of these numbers are known.

Content

1 Introductory remarks
2 Definition
3 Search for Whale numbers
4 Examples
5 For other reasons
6 Clusters of China
7 notes
8 Literature

Introductory remarks

To determine whether an n- digit number N is a Kit number, we construct a sequence of numbers similar to a sequence of Fibonacci numbers starting with n decimal digits of the number N. Then we continue the sequence, adding as the next member the sum of the previous n members. By definition, N is a Whale number if N is a member of the sequence under construction.

As an example, consider the 3-digit number N = 197. This number gives the sequence:

1 , 9 , 7 , 17, 33, 57, 107, 197, 361, ...

Since 197 is included in the sequence, 197 is the Whale number.

Definition

The Whale number is a positive integer N , which appears as a member of a sequence given by a linear recursive formula with initial terms defined by the digits of the number itself. If an n- digit number is given

N=\sum _{i=0}^{n-1}10^{i}{d_{i}},

{\ displaystyle N = \ sum _ {i = 0} ^ {n-1} 10 ^ {i} {d_ {i}},}

sequence $S_{N}$ ${\ displaystyle S_ {N}}$ formed from initial members $d_{n-1},d_{n-2},\ldots ,d_{1},d_{0}$ ${\ displaystyle d_ {n-1}, d_ {n-2}, \ ldots, d_ {1}, d_ {0}}$ and continues by members obtained as the sum of the previous n members. If the number N appears in the sequence $S_{N}$ ${\ displaystyle S_ {N}}$ , then N , they say that it is a Whale number. Single-valued Whale numbers have the Whale property trivially and are usually excluded from consideration.

Finding Whale Numbers

Infinitely or not, the number of the Whale is currently the subject of controversy. Whale numbers are rare and hard to find. They can be searched for by an exhaustive search, and a more efficient algorithm is not yet known ^[2] . According to Keith, on average it is expected $\textstyle {\frac {9}{10}}\log _{2}{10}\approx 2.99$ ${\ displaystyle \ textstyle {\ frac {9} {10}} \ log _ {2} {10} \ approx 2.99}$ Whale numbers between consecutive powers of 10 ^[3] . Known results support this assessment.

Examples

14 , 19 , 28 , 47 , 61 , 75 , 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171, 33445755, 44121607, 129572008, ^[4] 251133297.

For other reasons

Base numbers 12

11, 15, 1Ɛ, 22, 2 ᘔ, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, ᘔᘔ, ƐƐ, 125, 215, 24 ᘔ, 405, 42 ᘔ, 654, 80 ᘔ, 8 ᘔ 3, ᘔ 59, 1022, 1662, 2044, 3066, 4088, 4 ᘔ 1 ᘔ, 4 ᘔƐ1, 50 ᘔᘔ, 8538, Ɛ18Ɛ, 17256, 18671, 24 ᘔ 78, 4718Ɛ, 517Ɛᘔ, 157617, 1 ᘔ 265 ᘔ, 5 ᘔ 4074, 5 ᘔƐ140, 6Ɛ1449

Whale Clusters

A whale cluster is a number of whales, of which one is a multiple of the other. For example, (14, 28), (1104, 2208) and (31331, 62662, 93993). Possibly, there are only these three examples of Whale clusters ^[5] .

Notes

↑ Keith, 1987 , p. 41-42.
↑ Earls, Lichtblau, Weisstein .
↑ Keith Numbers (unopened) .
↑ Whale Numbers
↑ Copeland .

Literature

Mike Keith. Repfigit Numbers // Journal of Recreational Mathematics . - 1987. - T. 19 , no. 2 .
Keith Number (unopened) . MathWorld .
14 197 and other Keith Numbers (Neopr.) . Numberphile Brady Haran .

[_a62712e595b7f197-1] Keith, 1987 , p. 41-42.

[_aafac0f06feeb47f-2] Earls, Lichtblau, Weisstein .

[3] Keith Numbers (unopened) .

[OEIS-4] Whale Numbers

[_e4a959803d660ccd-5] Copeland .