A323823 Lexicographically first sequence starting with a(1) = 1, with no duplicate term, such that a(n) is the result of a self-additive linear combination of its own digits (concatenated sometimes into substrings).

1, 100, 101, 143, 182, 273, 364, 429, 455, 546, 637, 693, 728, 819, 924, 1010, 1020, 1030, 1040, 1050, 1060, 1070, 1080, 1090, 1233, 1370, 1371, 3288, 8833, 10000, 10001, 10100, 10110, 10120, 10130, 10140, 10150, 10160, 10170, 10180, 10190, 10200, 10210, 10220, 10230, 10240, 10250, 10260, 10270, 10280, 10290, 10300, 10310, 10320

Offset

1,2

Comments

A linear combination is an operation like a*u + b*v + c*w + d*x + ... = N. [We want only (+) signs here, thus the word "additive" in the definition.] The coefficients a, b, c, d,... and u, v, w, x,... are determined by the digits of a(n) itself, concatenated sometimes into substrings (no substring with a leading zero is allowed). No digit belongs to more than one substring and all digits are involved in the linear combination.

Some patterns are distinguishable:

a(1067) = 138614 --- 13 86 14

a(1068) = 148515 --- 14 85 15

a(1069) = 158416 --- 15 84 16

a(1070) = 168317 --- 16 83 17

a(1071) = 178218 --- 17 82 18

a(1072) = 188119 --- 18 81 19

a(1073) = 198020 --- 19 80 20

a(1074) = 207921 --- 20 79 21

a(1075) = 217822 --- 21 78 22

[pattern stops there]

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..1165

Example

a(1) = 1 belongs to the sequence as 1 = 1*1 [the digits appearing to the left of the (*) sign rebuild a(n); the same is true with the digits appearing to the right of the (*) sign];
10 does not belong to the sequence as 10 = 10*1 + 0*0, although being true, involves 5 digits instead of 4;
36 does not belong to the sequence as 36 = 3*6 + 6*3, although being true and involving 4 digits, doesn't respect the order in which the digits should appear in the linear combination;
a(2) = 100 belongs to the sequence as 100 = 10*10 + 0*0;
a(3) = 101 belongs to the sequence as 101 = 10*10 + 1*1;
a(4) = 143 belongs to the sequence as 143 = 14*1 + 3*43;
a(5) = 182 belongs to the sequence as 182 = 18*1 + 2*82;
etc.

Crossrefs

Cf. A323821 and A323822 for sequences dealing with the same idea of linear combination.

Sequence in context: A280825 A233345 A091254 * A085251 A178530 A063010

Adjacent sequences: A323820 A323821 A323822 * A323824 A323825 A323826

Keyword

base,nonn

Author

Eric Angelini and Jean-Marc Falcoz, Jan 30 2019

Status

approved