A239016 Numbers not larger than any rotation of their digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 126, 127, 128, 129, 132

Offset: 1

Author

M. F. Hasler, Mar 08 2014

Keywords

• nonn
• base

Comments

The numbers with nonincreasing digits, A009994, form a subsequence which first differs at a(73)=132 (not in A009994) from this one.

This sequence is a subsequence of A072544: numbers whose smallest decimal digit is also the initial digit. A072544(65)=121 is the first such number not in this sequence.

This criterion involving "rotation" is part of the characterization of Lyndon words, see e.g. A102659, A102660, A210584, A210585. All of these are subsequences of this sequence. For example, A102659 = A213969 intersect A239016.

Examples

			The number 10 is excluded from this sequence because its "rotation" 01 is smaller than the number itself.
The same is the case for any number whose first digit is not the smallest one: rotating a smaller digit to the front will always yield a smaller number, independently of the other digits. For this reason, all terms must be in A072544.
a(73)=132 is in the sequence because the nontrivial rotations of its digits are 321 and 213, both larger than 132.
		

Programs

PARI

is_A239016(n)=vecsort(d=digits(n))==d||!for(i=1,#d-1,n>[1,10^(#d-i)]*divrem(n,10^i)&&return)

Python

def ok(n):
    s = str(n)
    if "".join(sorted(s)) == s: return True
    return all(n <= int(s[i:] + s[:i]) for i in range(1, len(s)))
print(list(filter(ok, range(133)))) # Michael S. Branicky, Aug 21 2021