Numbers Within Numbers

As sometimes happens, the number associated with my diurnal age seems to contain nothing much of interest. Today the number was 26930 and I was temporarily stuck. However, I noticed something in Numbers Aplenty. See Figure 1.

Figure 1

The hexadecimal representation of 26930 is 6932. All the digits in the hexadecimal representation are contained within the decimal representation, each in the same proportion. I thought that this would be an interesting criterion to apply to the numbers from 10 to 40000. The numbers from 0 to 9 trivially satisfy the criterion. 

It turns out that there are only 90 decimal numbers that satisfy. This is the list (Permalink):

Hex     Decimal

35 --- 53

173 --- 371

391 --- 913

411 --- 1041

412 --- 1042

413 --- 1043

414 --- 1044

415 --- 1045

416 --- 1046

417 --- 1047

418 --- 1048

419 --- 1049

541 --- 1345

791 --- 1937

821 --- 2081

822 --- 2082

823 --- 2083

824 --- 2084

825 --- 2085

826 --- 2086

827 --- 2087

828 --- 2088

829 --- 2089

1004 --- 4100

1415 --- 5141

1524 --- 5412

1826 --- 6182

2008 --- 8200

2419 --- 9241

3012 --- 12306

4010 --- 16400

4011 --- 16401

4012 --- 16402

4013 --- 16403

4014 --- 16404

4015 --- 16405

4017 --- 16407

4018 --- 16408

4019 --- 16409

4167 --- 16743

4815 --- 18453

5021 --- 20513

5042 --- 20546

5221 --- 21025

6251 --- 25169

6528 --- 25896

6582 --- 25986

6702 --- 26370

6872 --- 26738

6921 --- 26913

6932 --- 26930

7221 --- 29217

7412 --- 29714

7830 --- 30768

7840 --- 30784

8020 --- 32800

8021 --- 32801

8022 --- 32802

8024 --- 32804

8025 --- 32805

8026 --- 32806

8027 --- 32807

8028 --- 32808

8029 --- 32809

8052 --- 32850

8335 --- 33589

8834 --- 34868

8840 --- 34880

8841 --- 34881

8842 --- 34882

8844 --- 34884

8845 --- 34885

8846 --- 34886

8847 --- 34887

8848 --- 34888

8849 --- 34889

9026 --- 36902

9056 --- 36950

9274 --- 37492

9830 --- 38960

9831 --- 38961

9832 --- 38962

9833 --- 38963

9834 --- 38964

9835 --- 38965

9837 --- 38967

9838 --- 38968

9839 --- 38969

9848 --- 38984

9982 --- 39298

The hexadecimal numbers will have the same or fewer digits than the decimal ones. Here is the list of decimal numbers on their own:

53, 371, 913, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048, 1049, 1345, 1937, 2081, 2082, 2083, 2084, 2085, 2086, 2087, 2088, 2089, 4100, 5141, 5412, 6182, 8200, 9241, 12306, 16400, 16401, 16402, 16403, 16404, 16405, 16407, 16408, 16409, 16743, 18453, 20513, 20546, 21025, 25169, 25896, 25986, 26370, 26738, 26913, 26930, 29217, 29714, 30768, 30784, 32800, 32801, 32802, 32804, 32805, 32806, 32807, 32808, 32809, 32850, 33589, 34868, 34880, 34881, 34882, 34884, 34885, 34886, 34887, 34888, 34889, 36902, 36950, 37492, 38960, 38961, 38962, 38963, 38964, 38965, 38967, 38968, 38969, 38984, 39298

The algorithm can  be easily modified to accommodate bases up to 36 and the range can be extended, let's say to 100,000. Here are the results for base 36 in the range up to 100,000 (Permalink):

Base 36 Decimal

41 --- 145

75 --- 257

185 --- 1589

213 --- 2631

247 --- 2743

268 --- 2816

291 --- 2917

340 --- 4032

573 --- 6735

659 --- 7965

680 --- 8064

814 --- 10408

850 --- 10548

896 --- 10698

1357 --- 50731

1504 --- 53140

1875 --- 57281

There are only 17 decimal numbers that satisfy in this range. These are listed below in decimal form only:

145, 257, 1589, 2631, 2743, 2816, 2917, 4032, 6735, 7965, 8064, 10408, 10548, 10698, 50731, 53140, 57281

If we wish to look at bases below 10 then the numbers will have the same or more digits and so the algorithm will need to be modified. We will require all the digits in base 10 format to be contained within the number in the lower base format and these digits need to be in the same proportion. Here are the results for base 8 in range from 8 to 40,000 (Permalink):

Base 8 Decimal

107 --- 71

1347 --- 743

1357 --- 751

2371 --- 1273

3165 --- 1653

4152 --- 2154

6130 --- 3160

6131 --- 3161

6132 --- 3162

6133 --- 3163

6134 --- 3164

6135 --- 3165

6136 --- 3166

6137 --- 3167

6232 --- 3226

12765 --- 5621

13620 --- 6032

14562 --- 6514

15713 --- 7115

15720 --- 7120

15721 --- 7121

15722 --- 7122

15723 --- 7123

15724 --- 7124

15726 --- 7126

15727 --- 7127

15764 --- 7156

17074 --- 7740

37621 --- 16273

52371 --- 21753

73560 --- 30576

75341 --- 31457

102345 --- 34021

102473 --- 34107

103462 --- 34610

104365 --- 35061

105436 --- 35614

106273 --- 36027

106347 --- 36071

There are only 39 numbers that satisfy. Here is the list in decimal format only:

71, 743, 751, 1273, 1653, 2154, 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167, 3226, 5621, 6032, 6514, 7115, 7120, 7121, 7122, 7123, 7124, 7126, 7127, 7156, 7740, 16273, 21753, 30576, 31457, 34021, 34107, 34610, 35061, 35614, 36027, 36071

Thus it can be seen that whenever a number seems boring, there is always something of interest that remains to be discovered and that property can often be generalised, as was done here, to find interesting properties of other numbers.

Getting back to comparing decimal numbers to their hexadecimal equivalents we can make conditions more stringent by requiring that the lengths of both numbers be the same so that the digits of both are simply permutations of the other. In the range up to 100,000, there are 24 numbers that qualify. These are (Permalink):

Hex Decimal

35 --- 53

173 --- 371

391 --- 913

1004 --- 4100

1415 --- 5141

1524 --- 5412

1826 --- 6182

2008 --- 8200

2419 --- 9241

12570 --- 75120

12571 --- 75121

12572 --- 75122

12573 --- 75123

12574 --- 75124

12575 --- 75125

12576 --- 75126

12577 --- 75127

12578 --- 75128

12579 --- 75129

12765 --- 75621

15086 --- 86150

16549 --- 91465

18197 --- 98711

18499 --- 99481

The numbers in decimal format only are:

53, 371, 913, 4100, 5141, 5412, 6182, 8200, 9241, 75120, 75121, 75122, 75123, 75124, 75125, 75126, 75127, 75128, 75129, 75621, 86150, 91465, 98711, 99481

Similarly in the range up to 100,000, there are only 16 numbers in base 8 format that have the same digits as their decimal equivalants (Permalink):

Base 8 Decimal

2371 --- 1273

3165 --- 1653

4152 --- 2154

6130 --- 3160

6131 --- 3161

6132 --- 3162

6133 --- 3163

6134 --- 3164

6135 --- 3165

6136 --- 3166

6137 --- 3167

6232 --- 3226

37621 --- 16273

52371 --- 21753

73560 --- 30576

75341 --- 31457

Here are the numbers in decimal format only:

1273, 1653, 2154, 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167, 3226, 16273, 21753, 30576, 31457

Lightweight mathematics of course but sometimes it's good just to have fun with numbers. The table below shows the number of decimal numbers whose value in another base is an anagram of its digits. The bases range from 4 to 28. For bases 2 and 3 and for 29 and beyond, there are no such numbers.

Base 26 is interesting because the ten decimal numbers that satisfy are consecutive and the hexadecimal numbers formed from them are similarly consecutive and formed simply by moving the leading 9 of the decimal number one place to the right. Thus 910 in decimal becomes 190 in hexadecimal, 911 becomes 191 and so on until 919 that becomes 199.

Base 26 Decimal

190 --- 910

191 --- 911

192 --- 912

193 --- 913

194 --- 914

195 --- 915

196 --- 916

197 --- 917

198 --- 918

199 --- 919

All this can be explored via this permalink.