A326640 -id:A326640 - cp's OEIS frontend

A309151 Lexicographically earliest infinite sequence of distinct terms starting with a(1) = 1 such that, for n>1, a(n) doesn't share any digit with the cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n).

1, 2, 3, 5, 4, 6, 7, 8, 9, 10, 11, 12, 14, 22, 20, 23, 24, 30, 13, 15, 16, 18, 40, 19, 17, 25, 21, 31, 27, 26, 28, 29, 32, 34, 39, 33, 35, 36, 41, 43, 37, 38, 47, 42, 44, 45, 46, 48, 49, 53, 50, 54, 52, 55, 60, 57, 58, 59, 56, 62, 63, 68, 51, 61, 65, 64, 66, 69, 67, 70, 71, 73, 83, 80, 74, 75, 72, 76, 77, 78, 79, 81, 87, 82

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jul 14 2019

Comments from N. J. A. Sloane, Jul 15 2019: (Start)
If we leave out the word "infinite", the "Lexicographically earliest sequence of distinct terms starting with a(1) = 1 such that, for n>1, a(n) doesn't share any digit with the cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n)" is the single-term sequence 1.
By computer, Eric Angelini and Jean-Marc Falcoz have shown that there is no earlier start than the 84-term sequence shown in the DATA section, that is,
D = 1, 2, 3, 5, 4, 6, 7, 8, 9, 10, 11, 12, 14, 22, 20, 23, 24, 30, 13, 15, 16, 18, 40, 19, 17, 25, 21, 31, 27, 26, 28, 29, 32, 34, 39, 33, 35, 36, 41, 43, 37, 38, 47, 42, 44, 45, 46, 48, 49, 53, 50, 54, 52, 55, 60, 57, 58, 59, 56, 62, 63, 68, 51, 61, 65, 64, 66, 69, 67, 70, 71, 73, 83, 80, 74, 75, 72, 76, 77, 78, 79, 81, 87, 82
The partial sums to this point are
S = 1, 3, 6, 11, 15, 21, 28, 36, 45, 55, 66, 78, 92, 114, 134, 157, 181, 211, 224, 239, 255, 273, 313, 332, 349, 374, 395, 426, 453, 479, 507, 536, 568, 602, 641, 674, 709, 745, 786, 829, 866, 904, 951, 993, 1037, 1082, 1128, 1176, 1225, 1278, 1328, 1382, 1434, 1489, 1549, 1606, 1664, 1723, 1779, 1841, 1904, 1972, 2023, 2084, 2149, 2213, 2279, 2348, 2415, 2485, 2556, 2629, 2712, 2792, 2866, 2941, 3013, 3089, 3166, 3244, 3323, 3404, 3491, 3573
To show that the 84 initial terms are as claimed, we must show that there is at least one infinite continuation satisfying the condition. For this, extend D by the terms 4204, 3334, 66666, 33334, 666666, 333334, 6666666, ... (compare A308900). As a result, S is extended by 7777, 11111, 77777, 111111, 777777, 1111111, 7777777, ..., and corresponding terms of D and S have no digits in common.
So the first 84 terms are correct.
It should be straightforward to use this method to show that the 1000 terms in the a-file are correct. This a-file could then be changed to a b-file. (End)

			The sequence starts with 1, 2, 3, 5, 4, 6, 7, 8, 9, 10, 11, 12, 14,...
We can't assign 4 to a(4) as the cumulative sum at that stage would be 10 and a cumulative sum ending in 0 cannot be extended by any integer without infringing the rules.
Thus we assign 5 to a(4) and 4 to a(5). We cannot assign 13 to a(13) as the cumulative sum would then be 91, with the digit 1 of 91 colliding with the digit 1 of 13. Thus a(13) = 14. Etc.

Carole Dubois, Table of n, a(n) for n = 1..5001
Carole Dubois, graph
Carole Dubois, graph for list of successive unauthorized sums
Jean-Marc Falcoz, Conjectured table of n, a(n) for n = 1..1000

Partial sums form A324773, A316914 (one digit is shared by the cumulative sum instead of zero digit here), A316915 (two digits shared), A326638 (three digits shared), A326639 (four digits shared), A326640 (five digits shared).

A326638 For any k, the cumulative sum a(1) + a(2) + a(3) + ... + a(k) shares at least three digits with a(k). Lexicographically first sequence of positive integers without duplicate terms having this property.

Original entry on oeis.org

100, 1000, 101, 103, 104, 135, 171, 128, 119, 120, 109, 121, 256, 147, 239, 163, 304, 370, 340, 450, 240, 200, 150, 250, 600, 160, 130, 700, 170, 131, 812, 380, 168, 693, 379, 300, 110, 102, 106, 151, 167, 107, 108, 140, 111, 105, 112, 114, 115, 117, 118, 113, 126, 129, 123, 124, 125, 137, 122, 149

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jul 15 2019

			Here are the first terms of the sequence:
100,1000,101,103,104,135,171,128,119,120,...
and here are the cumulative sums:
100,1100,1201,1304,1408,1543,1714,1842,1961,2081,...
If we align a(n) and its cumulative sum, we see that at least three digits are shared:
  100,1000, 101, 103, 104 ,135, 171, 128, 119, 120,...
  100,1100,1201,1304,1408,1543,1714,1842,1961,2081,...

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10001

Cf. A309151 (where no digit is shared by the cumulative sum), A316914 (where one digit is shared instead of three, by the cumulative sum), A316915 (two digits shared), A326639 (four digits shared), A326640 (five digits shared).

A326639 For any k, the cumulative sum a(1) + a(2) + a(3) + ... + a(k) shares at least four digits with a(k). Lexicographically first sequence of positive integers without duplicate terms having this property.

Original entry on oeis.org

1000, 10000, 1001, 1003, 1004, 1035, 1061, 1127, 1184, 1095, 1200, 2300, 2050, 2370, 1280, 1300, 1030, 3040, 1530, 3950, 4390, 4080, 5030, 1040, 6000, 1600, 3600, 7200, 2700, 8300, 3800, 1800, 1002, 1129, 1936, 1649, 1296, 1799, 1090, 1010, 1012, 1020, 1038, 1008, 1005, 1006, 1007, 1011, 1013, 1014, 1105, 1120, 1031

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jul 15 2019

			Here are the first terms of the sequence:
1000,10000,1001,1003,1004,1035,1061,1127,1184,...
and here are the cumulative sums:
1000,11000,12001,13004,14008,15043,16104,17231,18415,...
If we align a(n) and its cumulative sum, we see that at least four digits are shared:
1000,10000, 1001, 1003, 1004, 1035 ,1061, 1127, 1184,...
1000,11000,12001,13004,14008,15043,16104,17231,18415,...

Jean-Marc Falcoz, Table of n, a(n) for n = 1..5001

Cf. A309151 (no digit is shared by the cumulative sum instead of four digits here), A316914 (one digit is shared), A316915 (two digits shared), A326638 (three digits shared), A326640 (five digits shared).

cp's OEIS Frontend

A309151 Lexicographically earliest infinite sequence of distinct terms starting with a(1) = 1 such that, for n>1, a(n) doesn't share any digit with the cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n).

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A326638 For any k, the cumulative sum a(1) + a(2) + a(3) + ... + a(k) shares at least three digits with a(k). Lexicographically first sequence of positive integers without duplicate terms having this property.

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A326639 For any k, the cumulative sum a(1) + a(2) + a(3) + ... + a(k) shares at least four digits with a(k). Lexicographically first sequence of positive integers without duplicate terms having this property.

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