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

Author

Eric Angelini and Hans Havermann, Apr 11 2018

Keywords

• nonn
• base

Comments

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.

Examples

			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, ...
		

Links

• 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

Crossrefs

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.

Extensions

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.