A339803 -id:A339803 - cp's OEIS frontend

A357826 Base-10 weaker Skolem-Langford numbers.

231213, 312132, 12132003, 23121300, 23421314, 30023121, 31213200, 41312432, 1214230043, 1312432004, 2342131400, 2412134003, 3004312142, 3400324121, 4002342131, 4131243200, 4562342536, 4635243265, 5364235246, 5623425364, 6352432654, 6425324635, 14156742352637, 14167345236275

Offset: 1

Marc Morgenegg, Oct 14 2022

Self-describing numbers: between two digits "d" there are d digits.
a(n) has either 0 or 2 instances of any digit, hence even number of digits, and in fact the number of digits of a(n) == 0 or 2 or 6 (mod 8).
"weaker" means that when the smallest digit is x, all digits from x to the largest digit must be present.
The smallest digit x could be any value, but it turns out the biggest is x = 3 with 28 terms in total.
This sequence has 3390 terms. The largest term is 867315136875420024.
See A108116 for the "weak" variant with another constraint, and A132291 for the "strong" variant with more constraints.

			41312432 is a term since both 4's are separated by four digits, the 1's by one, the 3's by three, the 2's by two. Every digit from 1 to 4 is present.

Michael S. Branicky, Table of n, a(n) for n = 1..3390

Cf. base-10 Skolem-Langford numbers: A108116 (weak), A132291 (strong), A339803 (super weak).

def afull(): # SL() is in A108116
    alst = []
    for d in range(1, 11):
        for b in range(11-d):
            dset = ("0123456789")[b:b+d]
            s = [0 for _ in range(2*d)]
            for an in sorted(SL(dset, s)):
                alst.append(an)
    return sorted(alst)
print(afull()[:22]) # Michael S. Branicky, Oct 14 2022

More terms from David A. Corneth, Oct 14 2022

A339611 Pick any digit d; there are exactly d digits between d and the closest duplicate of d (either before or after) in the sequence.

Original entry on oeis.org

1, 2, 13, 20, 0, 3, 4, 5, 6, 7, 8, 41, 51, 61, 71, 81, 9, 12, 14, 21, 31, 49, 32, 54, 23, 62, 53, 17, 16, 15, 18, 37, 25, 324, 68, 27, 42, 36, 24, 371, 91, 28, 121, 34, 59, 38, 46, 52, 73, 29, 63, 45, 72, 84, 251, 213, 48, 93, 64, 131, 57, 69

Offset: 1

Eric Angelini and Carole Dubois, Dec 09 2020

This is the lexicographically earliest sequence of distinct nonnegative terms with this property.
From Edward Moody, Dec 16 2020: (Start)
The sequence is infinite because any potential earliest finite sequence F with the property (i.e., with no "unmatched" digits) can be extended to an infinite sequence. One way to do this is by adding terms consisting of sufficient copies of the string 12132003 to avoid duplication. In some cases the first additional term needs an additional prefix to preserve the property:
If F does not end with 1 or 2, no additional prefix is required, e.g. 1, 2, 13, 20, 0, 3, 12132003, 1213200312132003, ...;
If F ends with 2, but not 32, the next term can begin 131003, e.g. [...]2, 131003, 12132003, 1213200312132003, ...;
If F ends with 32, but not 2432, the next term can begin 1410014, e.g. [...]32, 1410014, 12132003, 1213200312132003, ...;
If F ends with 2432, the next term can begin 15120025, e.g. [...]2432, 15120025, 12132003, 1213200312132003, ...;
If F ends with 1, but not 121, the next term can begin 2132003, e.g. 1, 2, 13, 20, 0, 3, 4, 5, 6, 7, 8, 41, 51, 61, 71, 81, 2132003, 12132003, 1213200312132003, ...;
If F ends with 121, the next term can begin 31003, e.g. [...]121, 31003, 12132003, 1213200312132003, ...
(End)

			There is 1 digit between the first 1 [of a(1) = 1] and its closest duplicate [the 1 of a(3) = 13];
There are 2 digits between the first 2 [of a(2) = 2] and its closest duplicate [the 2 of a(4) = 20];
There are 3 digits between the first 3 [of a(3) = 13] and its closest duplicate [the 3 of a(6) = 3];
There is no digit between the first 0 [of a(4) = 20] and its closest duplicate [the 0 of a(5) = 0];
There is no contradiction when picking the 1 of a(13) = 51 as there is 1 digit between it and the 1 of a(12) = 41 and 1 digit between it and the 1 of a(14) = 61; etc.

Kevin Ryde, Table of n, a(n) for n = 1..10064
Kevin Ryde, Perl program generating terms of the sequence

Cf. A339803 (strings).

cp's OEIS Frontend

A357826 Base-10 weaker Skolem-Langford numbers.

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