A377911 -id:A377911 - cp's OEIS frontend

A302656 Replacing each term of this sequence S with its digitsum produces a new sequence S' such that S' and S share the same succession of digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 109, 18, 10, 17, 19, 89, 100, 27, 26, 36, 199999999999, 11, 16, 20, 15, 12, 24, 199, 45, 54, 63, 72, 81, 90, 108, 117, 126, 135, 29, 79, 299, 69, 39, 101, 13, 289, 144, 22, 14, 23, 31, 33, 21, 25, 110, 35, 1000, 9999999999, 28, 44, 38, 34, 48, 42, 49, 32, 200, 153, 43

Offset: 1

Eric Angelini and Hans Havermann, Apr 11 2018

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that doesn't lead to a contradiction.
There are huge jumps in this sequence. For example, here are three consecutive terms: a(96) = 41, a(97) = 2*10^111-1, a(98) = 234.
Records after a(97) are:
a(176) = 2*10^1111-1
a(396) = 2*10^11111-1
a(463) = 2*10^111111-1
a(1918) = 2*10^1111111-1
...
It seems likely that every number will eventually appear (see A376772). After 1262743 terms, according to Dominic McCarty, the smallest missing number is 387 = 9*43. - N. J. A. Sloane, Nov 24 2024
Comment from N. J. A. Sloane, Dec 14 2024 (Start)
Dominic McCarty reports that he has computed 3026560 terms (easy to remember). The only headway he has made is that if any string of digits d1, d2, ..., dn appears in the digit stream with density > 0, then it can be shown that all numbers whose digitsum is the concatenation of d1, d2, ..., dn will eventually appear. (End)
The OEIS contains 16 sequences derived from the present one, none of which seem to have appeared in any other context: see A376769-A376776, A377903-A377904, A377906-A377911.

			The first nine terms do not change when replaced by their digitsum;
109 = a(10) is replaced by the digitsum 1 + 0 + 9 = 10;
18 = a(11) is replaced by the digitsum 1 + 8 = 9;
10 = a(12) is replaced by the digitsum 1 + 0 = 1;
17 = a(13) is replaced by the digitsum 1 + 7 = 8;
19 = a(14) is replaced by the digitsum 1 + 9 = 10;
89 = a(15) is replaced by the digitsum 8 + 9 = 17;
100 = a(16) is replaced by the digitsum 1 + 0 + 0 = 1;
27 = a(17) is replaced by the digitsum 2 + 7 = 9;
26 = a(18) is replaced by the digitsum 2 + 6 = 8;
36 = a(19) is replaced by the digitsum 3 + 6 = 9;
199999999999 = a(20) is replaced by the digitsum 1 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 100; etc.
We see that the first and the last column here (the terms of S, which is the present sequence, and S', which is A376769) share the same succession of digits (A376771):
1, 0, 9, 1, 8, 1, 0, 1, 7, 1, 9, 8, 9, 1, 0, 0, 2, 7, 2, 6, 3, 6, 1, 9, 9, 9, 9, ...

Michael S. Branicky, Table of n, a(n) for n = 1..175 (matches terms computed by Hans Havermann)
Michael S. Branicky, Table of n, a(n) for n = 1..1982 (with terms > 1000 digits summarized; corrects terms in Hans Havermann data starting from a(312))
Michael S. Branicky, Python program for A302656
Dominic McCarty, C++ program for A302656
Dominic McCarty, Log scatterplot of A302656 (terms greater than 10^25 omitted)
Dominic McCarty, Table of n, a(n), digsum(a(n)) for n = 1..10000

Cf. A007953 (digitsum of n), A376769 (digitsum of a(n)), A376770-A376774.
For records, see A377903 and A377904.
Summary: the 16 sequences derived from the present one are A376769-A376776, A377903-A377904, A377906-A377911.

Michael S. Branicky noticed that there were errors in Hans Havermann's data. Following his advice, I deleted Hans's incorrect 2279-term data file and a graph that was based on it. - N. J. A. Sloane, Nov 05 2024.

A376769 a(n) = digitsum of A302656(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 1, 8, 10, 17, 1, 9, 8, 9, 100, 2, 7, 2, 6, 3, 6, 19, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 16, 20, 15, 12, 2, 4, 19, 9, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 90, 10, 8, 11, 7, 12, 6, 13, 5, 2, 9, 7, 9, 2, 9, 9, 6, 9, 3, 9, 10, 11, 3, 2, 8, 9, 14, 4, 22, 14, 23, 3, 13, 3, 21, 25, 1, 10, 3, 5, 1000, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 8, 4

Offset: 1

N. J. A. Sloane, Nov 04 2024

Of course, by the definition of A302656, the present sequence and A302656 have exactly the same sequence of digits (which is now A376771).

Michael S. Branicky, Table of n, a(n) for n = 1..1982

Cf. A007953, A302656, A376770, A376771.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A376776 Position of prime(n) in A302656, or -1 if prime(n) does not appear in A302656.

Original entry on oeis.org

2, 3, 5, 7, 21, 44, 13, 14, 49, 38, 50, 77, 96, 68, 78, 81, 83, 138, 401, 113, 149, 39, 181, 15, 1646, 43, 110, 161, 10, 153, 364, 162, 432, 1679, 471, 451, 1683, 425, 1691, 1615, 13649

Offset: 1

N. J. A. Sloane, Nov 06 2024

The primes in A302656 do not appear in their natural order.

			Prime(5) = 11, and A302656(21) = 11, so a(5) = 11.
Prime(40) = 173, and A302656(1615) = 173, so a(40) = 1615.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(41) = 13649 from Dominic McCarty's b-file for A376772 added by N. J. A. Sloane, Nov 21 2024

A377903 Indices of records in A302656.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 97, 176, 396, 463, 1918, 1984, 2278

Offset: 1

N. J. A. Sloane, Nov 19 2024

Conjecture: a(n) = A377904(n-9) for n >= 11.

Based on Dominic McCarty's data in A302656.

			Table showing n, a(n) = index of n-th record in A302656, and the corresponding record value:
   1 1 1
   2 2 2
   3 3 3
   4 4 4
   5 5 5
   6 6 6
   7 7 7
   8 8 8
   9 9 9
   10 10 109
   11 20 199999999999
   12 97 2*10^111-1
   13 176 2*10^1111-1
   14 396 2*10^11111-1
   15 463 2*10^111111-1
   16 1918 2*10^1111111-1
   17 1984 2*10^11111111-1
   18 2278 2*10^111111111-1

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A377904 Index of 2*10^((10^n-1)/9) - 1 in A302656, or -1 if that number never appears there.

Original entry on oeis.org

1, 14, 20, 97, 176, 396, 463, 1918, 1984, 2278

Offset: 0

N. J. A. Sloane, Nov 19 2024

a(n) is the index in A302656 where we see 199...9, with k 9's, where k = 11...1 (with n 1's). For example, a(3) = 97 is where we see 199..9 with 111 9's.
It is extremely unlikely that -1 appears as a term in this sequence.
Based on Dominic McCarty's data in A302656.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A377906 Index of 10^n in A302656, or -1 if 10^n does not appear in A302656.

Original entry on oeis.org

1, 12, 16, 56, 93, 136, 168, 321, 332, 363, 409, 411, 443, 467, 1658, 1688, 1699, 1708, 1715, 1720, 1913

Offset: 0

N. J. A. Sloane, Nov 22 2024

Also the indices of the 1's in A376769.
A term 10^n in A302656 is the trigger for an astronomically-sized term 2*10^{n 1's} - 1 later. For example, a(4) = 93 leads to the term A302656(176) = 2*10^1111 - 1  (cf. A377903, A377904).
a(0) through a(19) based on Dominic McCarty's data in A302656.
The initial 21 terms form three groups of sizes 7, 7, and at least 7. It would be nice to know more terms.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A376771 List of successive digits (the "digit stream") of A302656 (or, equally, A376769).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 9, 1, 8, 1, 0, 1, 7, 1, 9, 8, 9, 1, 0, 0, 2, 7, 2, 6, 3, 6, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 1, 1, 1, 6, 2, 0, 1, 5, 1, 2, 2, 4, 1, 9, 9, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 1, 0, 8, 1, 1, 7, 1, 2, 6, 1, 3, 5, 2, 9, 7, 9, 2, 9, 9, 6, 9, 3, 9, 1, 0, 1, 1, 3, 2, 8, 9, 1, 4, 4, 2, 2, 1, 4, 2, 3, 3, 1, 3, 3, 2, 1, 2, 5, 1, 1, 0, 3, 5, 1

Offset: 1

N. J. A. Sloane, Nov 04 2024

N. J. A. Sloane, Table of n, a(n) for n = 1..2361

Cf. A302656, A376769.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A376772 Index where n appears in A302656, or -1 if n does not appear there.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 21, 25, 44, 48, 24, 22, 13, 11, 14, 23, 52, 47, 49, 26, 53, 18, 17, 58, 38, 75, 50, 65, 51, 61, 55, 19, 77, 60, 42, 84, 96, 63, 68, 59, 28, 94, 78, 62, 64, 135, 73, 132, 81, 29, 146, 131, 300, 89, 83, 141, 138, 109, 30, 148, 133, 317, 401, 86, 41, 143, 113, 31, 149, 165, 323, 452, 180, 124, 39, 117, 32, 315, 181, 369, 478, 447, 166, 128, 15, 33

Offset: 1

N. J. A. Sloane, Nov 04 2024; updated Nov 21 2024

nonn

387 doesn't appear for n < 1262743. - Dominic McCarty, Nov 07 2024

Dominic McCarty, Table of n, a(n) for n = 1..386 (terms 1..178 from N. J. A. Sloane)

Cf. A302656, A376776.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A376770 a(n) = digitsum of A376769(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 9, 1, 8, 1, 8, 1, 9, 8, 9, 1, 2, 7, 2, 6, 3, 6, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 7, 2, 6, 3, 2, 4, 10, 9, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 1, 8, 2, 7, 3, 6, 4, 5, 2, 9, 7, 9, 2, 9, 9, 6, 9, 3, 9, 1, 2, 3, 2, 8, 9, 5, 4, 4, 5, 5, 3, 4, 3, 3, 7, 1, 1, 3, 5, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 8, 4

Offset: 1

N. J. A. Sloane, Nov 04 2024

Michael S. Branicky, Table of n, a(n) for n = 1..1982

Cf. A007953, A302656, A376769.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

A376773 Records in A376772.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 21, 25, 44, 48, 52, 53, 58, 75, 77, 84, 96, 135, 146, 300, 317, 401, 452, 478, 1608, 1677, 1679, 1681, 1683, 1703, 1753, 1773, 13649, 13704, 124912, 124925, 125336, 128212, 128221, 128347, 128376, 128529

Offset: 1

N. J. A. Sloane, Nov 05 2024 (with thanks to Michael S. Branicky)

Numbers that are the slowest to appear in A302656.

Cf. A302656, A376772, A376774, A376775.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(37)-a(46) from Dominic McCarty, Nov 08 2024

cp's OEIS Frontend

A302656 Replacing each term of this sequence S with its digitsum produces a new sequence S' such that S' and S share the same succession of digits.

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A376769 a(n) = digitsum of A302656(n).

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A376776 Position of prime(n) in A302656, or -1 if prime(n) does not appear in A302656.

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A377903 Indices of records in A302656.

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A377904 Index of 2*10^((10^n-1)/9) - 1 in A302656, or -1 if that number never appears there.

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A377906 Index of 10^n in A302656, or -1 if 10^n does not appear in A302656.

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A376771 List of successive digits (the "digit stream") of A302656 (or, equally, A376769).

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A376772 Index where n appears in A302656, or -1 if n does not appear there.

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A376770 a(n) = digitsum of A376769(n).

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A376773 Records in A376772.

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