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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.

%I A302656 #87 Dec 14 2024 00:53:01
%S A302656 1,2,3,4,5,6,7,8,9,109,18,10,17,19,89,100,27,26,36,199999999999,11,16,
%T A302656 20,15,12,24,199,45,54,63,72,81,90,108,117,126,135,29,79,299,69,39,
%U A302656 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
%N 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.
%C A302656 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.
%C A302656 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.
%C A302656 Records after a(97) are:
%C A302656 a(176) = 2*10^1111-1
%C A302656 a(396) = 2*10^11111-1
%C A302656 a(463) = 2*10^111111-1
%C A302656 a(1918) = 2*10^1111111-1
%C A302656 ...
%C A302656 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
%C A302656 Comment from _N. J. A. Sloane_, Dec 14 2024 (Start)
%C A302656 _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)
%C A302656 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.
%H A302656 Michael S. Branicky, <a href="/A302656/b302656.txt">Table of n, a(n) for n = 1..175</a> (matches terms computed by Hans Havermann)
%H A302656 Michael S. Branicky, <a href="/A302656/a302656_1.txt">Table of n, a(n) for n = 1..1982</a> (with terms > 1000 digits summarized; corrects terms in Hans Havermann data starting from a(312))
%H A302656 Michael S. Branicky, <a href="/A302656/a302656_2.txt">Python program for A302656</a>
%H A302656 Dominic McCarty, <a href="/A302656/a302656_3.txt">C++ program for A302656</a>
%H A302656 Dominic McCarty, <a href="/A302656/a302656_2.png">Log scatterplot of A302656</a> (terms greater than 10^25 omitted)
%H A302656 Dominic McCarty, <a href="/A302656/a302656_6.txt">Table of n, a(n), digsum(a(n)) for n = 1..10000</a>
%e A302656 The first nine terms do not change when replaced by their digitsum;
%e A302656 109 = a(10) is replaced by the digitsum 1 + 0 + 9 = 10;
%e A302656 18 = a(11) is replaced by the digitsum 1 + 8 = 9;
%e A302656 10 = a(12) is replaced by the digitsum 1 + 0 = 1;
%e A302656 17 = a(13) is replaced by the digitsum 1 + 7 = 8;
%e A302656 19 = a(14) is replaced by the digitsum 1 + 9 = 10;
%e A302656 89 = a(15) is replaced by the digitsum 8 + 9 = 17;
%e A302656 100 = a(16) is replaced by the digitsum 1 + 0 + 0 = 1;
%e A302656 27 = a(17) is replaced by the digitsum 2 + 7 = 9;
%e A302656 26 = a(18) is replaced by the digitsum 2 + 6 = 8;
%e A302656 36 = a(19) is replaced by the digitsum 3 + 6 = 9;
%e A302656 199999999999 = a(20) is replaced by the digitsum 1 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 100; etc.
%e A302656 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):
%e A302656 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, ...
%Y A302656 Cf. A007953 (digitsum of n), A376769 (digitsum of a(n)), A376770-A376774.
%Y A302656 For records, see A377903 and A377904.
%Y A302656 Summary: the 16 sequences derived from the present one are A376769-A376776, A377903-A377904, A377906-A377911.
%K A302656 nonn,base
%O A302656 1,2
%A A302656 _Eric Angelini_ and _Hans Havermann_, Apr 11 2018
%E A302656 _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.