A302502 Lexicographically first sequence of distinct terms such that any set of seven successive digits can be reordered as {d, d+1, d+2, d+3, d+4, d+5, d+6}, d being the smallest of the seven digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 56, 78, 23, 45, 67, 12, 345, 60, 123, 456, 71, 234, 560, 1234, 567, 82, 3456, 712, 34560, 12345, 601, 2345, 671, 23456, 782, 34567, 89, 345678, 93, 4567, 823, 45671, 234560, 123456, 789, 3456782, 345671, 234567, 893, 45678, 934, 5678, 9345, 678, 93456, 7823, 456712, 345601

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Apr 09 2018

Keywords

• nonn
• base

Comments

As the digit 0 has no predecessor and the digit 9 has no successor here, sets of successive digits like {5,4,3,2,1,0,9} and {4,5,6,7,8,9,0} are forbidden.

Examples

			Terms a(1) to a(10) are obvious;
a(11) is 34 because 34 is the smallest integer not yet in the sequence such that the elements of the sets {4,5,6,7,8,9,3} and {5,6,7,8,9,3,4} are seven consecutive digits;
a(12) is 56 because 56 is the smallest integer not yet in the sequence such that the elements of the sets {6,7,8,9,3,4,5} and {7,8,9,3,4,5,6} are seven consecutive digits;
a(13) is 78 because 78 is the smallest integer not yet in the sequence such that the elements of the sets {8,9,3,4,5,6,7} and {9,3,4,5,6,7,8} are seven consecutive digits;
etc.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A228326 for the same idea with sets of two digits, A302173 (sets of three digits), A302499 (sets of four digits), A302500 (sets of five digits) and A302501 (sets of six digits).

Programs

Python

a, runLength = [i for i in range(10)], 7
def helper(s, k, l, a):
    if k not in a: return k
    return min([helper(s[(2-l):]+str(i), int(str(k)+str(i)), l, a) for i in range(10) if (k!=0 or i!=0) and s.find(str(i))==-1 and (all(d[n]+1==d[n+1] for n in range(l-1)) if (d:=sorted([int((s+str(i))[n]) for n in range(l)])) else False)])
while len(a)<100: a.append(helper(("".join(map(str, a)))[(1-runLength):], 0, runLength, a))
print(a) # Dominic McCarty, Feb 03 2025