In my post titled Additive Fibonacci-like Numbers I was dealing with additive digital roots to generate additional digits after the starting two digits were in place. For example, let's start with 78:$$ 78 \rightarrow 7 + 8 =15 \rightarrow 1+5=6$$This gives 786 and now:$$86 \rightarrow 8+6=14 \rightarrow 1+4=5$$Thus the new number is 7865 and so in summary:$$78 \rightarrow 786 \rightarrow 7865$$The advantage of this approach is that the sum of the two previous digits reduces to a single digit between 1 and 9. Let's call these types of numbers Additive Fibonacci-like Numbers of the First Type. Between 100 and 1,000,000 these numbers are:
101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999
However, in my previous post, Variations on the Taxi Cab Number, I was not working with the digital roots and this is a severe limitation. The early digits need to be small if the digits are to progress in a Fibonacci-like manner. That's why, in the range of numbers, up to one million, the largest number is 303369. Let's call these types of numbers Additive Fibonacci-like Numbers of the Second Type. Between 100 and 1,000,000 these numbers are:
101, 112, 123, 134, 145, 156, 167, 178, 189, 202, 213, 224, 235, 246, 257, 268, 279, 303, 314, 325, 336, 347, 358, 369, 404, 415, 426, 437, 448, 459, 505, 516, 527, 538, 549, 606, 617, 628, 639, 707, 718, 729, 808, 819, 909, 1011, 1123, 1235, 1347, 1459, 2022, 2134, 2246, 2358, 3033, 3145, 3257, 3369, 4044, 4156, 4268, 5055, 5167, 5279, 6066, 6178, 7077, 7189, 8088, 9099, 10112, 11235, 12358, 20224, 21347, 30336, 31459, 40448, 101123, 112358, 202246, 303369
With bases higher than 10, this limitation is reduced. For example in base 16, if we start as before with an initial 78 then a third digit is possible:$$78 \rightarrow 7 + 8 = 15 = F$$Thus we have:$$ \begin{align} 78F_{16} &= 7 \times 16^2 + 8 \times 16 + 15 \\ &=1935_{10} \end{align} $$This means that 1935 is an Additive Fibonacci-like Number of the Second Type in base 16. Here is a list of numbers greater than 27000 and less than 40000 that are "additive Fibonacci-like" and of the "second type" in base 16 (permalink):
- 28791 --> 7077
- 29065 --> 7189
- 29339 --> 729b
- 29613 --> 73ad
- 29887 --> 74bf
- 32904 --> 8088
- 33178 --> 819a
- 33452 --> 82ac
- 33726 --> 83be
- 37017 --> 9099
- 37291 --> 91ab
- 37565 --> 92bd
- 37839 --> 93cf
Additive Fibonacci-like Numbers of the Second Type in base 16 are:
28791, 29065, 29339, 29613, 29887, 32904, 33178, 33452, 33726, 37017, 37291, 37565, 37839
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Here the numbers greater than 27000 and less than 40000 for base 15:
- 27128 --> 8088
- 27370 --> 819a
- 27612 --> 82ac
- 27854 --> 83be
- 30519 --> 9099
- 30761 --> 91ab
- 31003 --> 92bd
- 33910 --> a0aa
- 34152 --> a1bc
- 34394 --> a2ce
- 37301 --> b0bb
- 37543 --> b1cd
Additive Fibonacci-like Numbers of the Second Type in base 15 are (permalink):
27128, 27370, 27612, 27854, 30519, 30761, 31003, 33910, 34152, 34394, 37301, 37543
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Here are the numbers greater than 27000 and less than 40000 for base 14 (permalink):
- 27590 --> a0aa
- 27802 --> a1bc
- 30349 --> b0bb
- 30561 --> b1cd
- 33108 --> c0cc
- 35867 --> d0dd
- 38628 --> 10112
Additive Fibonacci-like Numbers of the Second Type in base 14 are:
27590, 27802, 30349, 30561, 33108, 35867, 38628
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Here are the numbers greater than 27000 and less than 40000 for base 13 (permalink):
- 28745 --> 10112
- 31140 --> 11235
- 33535 --> 12358
- 35930 --> 1347b
Additive Fibonacci-like Numbers of the Second Type in base 13 are:
28745, 31140, 33535, 35930
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For base 12, there are none between 27000 and 40000 but for base 11 we have (permalink):
- 29550 --> 20224
- 31027 --> 21347
- 32504 --> 2246a
Additive Fibonacci-like Numbers of the Second Type in base 11 are:
29550, 31027, 32504
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For base 10, we have (permalink):
- 30336 --> 30336
- 31459 --> 31459
Additive Fibonacci-like Numbers of the Second Type in base 10 are:
30336, 31459
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For base 9 there are none but for base 8 there is one (permalink):
- 33363 --> 101123
Additive Fibonacci-like Numbers of the Second Type in base 8 are:
33363
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For base 7, we have
- 34432 --> 202246
Additive Fibonacci-like Numbers of the Second Type in base 7 are:
34432
There are none for bases 2, 3, 4, 5 and 6. I've added this determination of whether a number is additive Fibonacci-like of the second type to my multipurpose algorithm.
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