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PrimeFan's Listing of Esoteric Integer Sequences

Main | Digit Addition Generator Chains

Chains of Digit Addition Generators

These sequences are of course infinite, but for my convenience, I only plan to go up to about 1000 in each chain, and up to the self number closest to 200.

The initial terms of the following sequences correspond to the self numbers (Sloane's A003052).

DAGCH00001

Sequence

1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101, 103, 107, 115, 122, 127, 137, 148, 161, 169, 185, 199, 218, 229, 242, 250, 257, 271, 281, 292, 305, 313, 320, 325, 335, 346, 359, 376, 392, 406, 416, 427, 440, 448, 464, 478, 497, 517, 530, 538, 554, 568, 587, 607, 620, 628, 644, 658, 677, 697, 719, 736, 752, 766, 785, 805, 818, 835, 851, 865, 884, 904, 917, 934, 950, 964, 983, 1003

Description

Chain of digit addition generators with initial term 1. After initial term, each term is equal to the previous term plus the sum of its digits.

Mathematica Command

searchMax = 1000; dagChain1 = {1}; For[n = 1, dagChain1[[n]] < searchMax, n++, dagResult = dagChain1[[n]] + (Plus @@ IntegerDigits[dagChain1[[n]]]); dagChain1 = Flatten[{dagChain1, dagResult}]]; dagChain1

Thanks to Alonso Delarte for this program. He writes that this program can easily be modified for any other digit addition generator chain, and in fact, I have used it for each of the following chains.

Comments

This sequence is Sloane's A004207.


DAGCH00003

Sequence

3, 6, 12, 15, 21, 24, 30, 33, 39, 51, 57, 69, 84, 96, 111, 114, 120, 123, 129, 141, 147, 159, 174, 186, 201, 204, 210, 213, 219, 231, 237, 249, 264, 276, 291, 303, 309, 321, 327, 339, 354, 366, 381, 393, 408, 420, 426, 438, 453, 465, 480, 492, 507, 519, 534, 546, 561, 573, 588, 609, 624, 636, 651, 663, 678, 699, 723, 735, 750, 762, 777, 798, 822, 834, 849, 870, 885, 906, 921, 933, 948, 969, 993, 1014

Description

Chain of digit addition generators with initial term 3. After initial term, each term is equal to the previous term plus the sum of its digits. A little reflection will show that every term of this sequence must be a multiple of 3.

Comments

Sloane's A016052.


DAGCH00005

Sequence

5, 10, 11, 13, 17, 25, 32, 37, 47, 58, 71, 79, 95, 109, 119, 130, 134, 142, 149, 163, 173, 184, 197, 214, 221, 226, 236, 247, 260, 268, 284, 298, 317, 328, 341, 349, 365, 379, 398, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620, 628, 644, 658, 677, 697, 719, 736, 752, 766, 785, 805, 818, 835, 851, 865, 884, 904, 917, 934, 950, 964, 983, 1003

Description

Chain of digit addition generators with initial term 5. After initial term, each term is equal to the previous term plus the sum of its digits.

Comments

Sloane's A007618.


DAGCH00007

Sequence

7, 14, 19, 29, 40, 44, 52, 59, 73, 83, 94, 107

Description

Chain of digit addition generators with initial term 7. After initial term, each term is equal to the previous term plus the sum of its digits. At 107, this sequence converges with DAGCH00001.

Comments

Sloane's A006507.


DAGCH00009

Sequence

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 117, 126, 135, 144, 153, 162, 171, 180, 189, 207, 216, 225, 234, 243, 252, 261, 270, 279, 297, 315, 324, 333, 342, 351, 360, 369, 387, 405, 414, 423, 432, 441, 450, 459, 477, 495, 513, 522, 531, 540, 549, 567, 585, 603, 612, 621, 630, 639, 657, 675, 693, 711, 720, 729, 747, 765, 783, 801, 810, 819, 837, 855, 873, 891, 909, 927, 945, 963, 981, 999, 1026

Description

Chain of digit addition generators with initial term 9. After initial term, each term is equal to the previous term plus the sum of its digits. A little reflection will show that every term of this sequence must be a multiple of 9. As a matter of fact, you have to go the 12th term to verify that this sequence is in fact not equal to the sequence of multiples of 9.

Comments

Sloane's A016096.


DAGCH00020

Sequence

20, 22, 26, 34, 41, 46, 56, 67, 80, 88, 104, 109

Description

Chain of digit addition generators with initial term 20. After initial term, each term is equal to the previous term plus the sum of its digits. At 109, this sequence converges with DAGCH00005.

Comments

Sloane's A036227.


DAGCH00053

Sequence

53, 61, 68, 82, 92, 103

Description

Chain of digit addition generators with initial term 53. After initial term, each term is equal to the previous term plus the sum of its digits. At 103, this sequence converges with DAGCH00001.


DAGCH00064

Sequence

64, 74, 85, 98, 115

Description

Chain of digit addition generators with initial term 64. After initial term, each term is equal to the previous term plus the sum of its digits. At 115, this sequence converges with DAGCH00001.


DAGCH00075

Sequence

75, 87, 102, 105, 111

Description

Chain of digit addition generators with initial term 75. After initial term, each term is equal to the previous term plus the sum of its digits. At 111, this sequence converges with DAGCH00003.


DAGCH00086

Sequence

86, 100, 101

Description

Chain of digit addition generators with initial term 86. After initial term, each term is equal to the previous term plus the sum of its digits. At 101, this sequence converges with DAGCH00001.


DAGCH00097

Sequence

97, 113, 118, 128, 139, 152, 160, 167, 181, 191, 202, 206, 214, 221, 226, 236, 247, 260, 268, 284, 298, 317, 328, 341, 349, 365, 379, 398, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620

Description

Chain of digit addition generators with initial term 64. After initial term, each term is equal to the previous term plus the sum of its digits. At 115, this sequence converges with DAGCH00001.


DAGCH00108

Sequence

108, 117

Description

Chain of digit addition generators with initial term 108. After initial term, each term is equal to the previous term plus the sum of its digits. At 117, this sequence converges with DAGCH00009.


DAGCH00110

Sequence

110, 112, 116, 124, 131, 136, 146, 157, 170, 178, 194, 208, 218

Description

Chain of digit addition generators with initial term 110. After initial term, each term is equal to the previous term plus the sum of its digits. At 218, this sequence converges with DAGCH00001.

Comments

Sloane's A036230.


DAGCH00121

Sequence

121, 125, 133, 140, 145, 155, 166, 179, 196, 212, 217, 227, 238, 251, 259, 275, 289, 308, 319, 332, 340, 347, 361, 371, 382, 395, 412, 419, 433, 443, 454, 467, 484, 500, 505, 515, 526, 539, 556, 572, 586, 605, 616, 629, 646, 662, 676, 695, 715, 728, 745, 761, 775, 794, 814, 827, 844, 860, 874, 893, 913, 926, 943, 959, 982, 1001

Description

Chain of digit addition generators with initial term 121. After initial term, each term is equal to the previous term plus the sum of its digits.

Comments

Sloane's A036231.


DAGCH00132

Sequence

132, 138, 150, 156, 168, 183, 195, 210

Description

Chain of digit addition generators with initial term 132. After initial term, each term is equal to the previous term plus the sum of its digits. At 210, this sequence converges with DAGCH00003.


DAGCH00143

Sequence

143, 151, 158, 172, 182, 193, 206

Description

Chain of digit addition generators with initial term 143. After initial term, each term is equal to the previous term plus the sum of its digits. At 206, this sequence converges with DAGCH00097, which in turn converges with DAGCH00001 at 620.


DAGCH00154

Sequence

154, 164, 175, 188, 205, 212

Description

Chain of digit addition generators with initial term 154. After initial term, each term is equal to the previous term plus the sum of its digits. At 212, this sequence converges with DAGCH00121.


DAGCH00165

Sequence

165, 177, 192, 204

Description

Chain of digit addition generators with initial term 165. After initial term, each term is equal to the previous term plus the sum of its digits. At 204, this sequence converges with DAGCH00003.


DAGCH00176

Sequence

176, 190, 200, 202

Description

Chain of digit addition generators with initial term 176. After initial term, each term is equal to the previous term plus the sum of its digits. At 202, this sequence converges with DAGCH00097.

Comments

Coincidentally, the four terms given here appear in Sloane's A067943, a sequence involving repunits and modular arithmetic. The term after 202 in Sloane's A067943 is 205, differing from the next term 206 in this one which matches DAGCH00097.


DAGCH00187

Sequence

187, 203, 208, 218

Description

Chain of digit addition generators with initial term 187. After initial term, each term is equal to the previous term plus the sum of its digits. At 218, this sequence converges with DAGCH00001.


DAGCH00198

Sequence

198, 216

Description

Chain of digit addition generators with initial term 198. After initial term, each term is equal to the previous term plus the sum of its digits. At 216, this sequence converges with DAGCH00009.

Comments

Coincidentally, terms from 198 to 297 in this sequence are in Sloane's A045776, a(n + 1) is smallest multiple of (sum of digits of a(n)) which is > a(n). Given that these are both base-dependent sequences involving sums of digits, the coincidence is enticing.


DAGCH00209

Sequence

209, 220, 224, 232, 239, 253, 263, 274, 287, 304, 311, 316, 326, 337, 350, 358, 374, 388, 407, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620

Description

Chain of digit addition generators with initial term 209. After initial term, each term is equal to the previous term plus the sum of its digits. At 620, this sequence converges with DAGCH00001.


DAGCH00211

Sequence

211, 215, 223, 230, 235, 245, 256, 269, 286, 302, 307, 317, 328, 341, 349, 365, 379, 398, 418

Description

Chain of digit addition generators with initial term 211. After initial term, each term is equal to the previous term plus the sum of its digits. At 418, this sequence converges with DAGCH00209.