A354049 The smallest number that includes all the digits of n but does not equal n.

10, 10, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 21, 31, 41, 51, 61, 71, 81, 91, 102, 12, 122, 32, 42, 52, 62, 72, 82, 92, 103, 13, 23, 133, 43, 53, 63, 73, 83, 93, 104, 14, 24, 34, 144, 54, 64, 74, 84, 94, 105, 15, 25, 35, 45, 155, 65, 75, 85, 95, 106, 16, 26, 36, 46, 56, 166, 76, 86, 96, 107

Offset: 0

Author

Scott R. Shannon and Zach J. Shannon, May 16 2022

Keywords

• nonn
• base
• look

Comments

The terms cannot start with a leading zero so any number including a zero must have at least one digit greater than zero as its first digit. See the examples below.

Examples

			a(9) = 19 as there is no smaller number that includes the digit 9 but does not equal 9.
a(10) = 100 as there is no smaller number that includes the digits 1 and 0 but does not equal 10. Note that '01' = 1 is not allowed.
a(20) = 102 as there is no smaller number that includes the digits 2 and 0 but does not equal 20. Note that '02' = 2 is not allowed.
a(22) = 122 as there is no smaller number that includes two 2 digits but does not equal 22.
a(200) = 1002 as there is no smaller number that includes two 0 digits and the digit 2 but does not equal 200.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 0..9999
• Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Crossrefs

Cf. A004185, A004185, A337227.

Programs

PARI

vd(n) = my(d=if (n, digits(n), [0])); vector(10, k, #select(x->(x==k-1), d));
isok(k, n, d) = if (k!=n, my(dd=vd(k)); for (i=1, #d, if (dd[i] < d[i], return(0))); return(1));
a(n) = my(k=0, d=vd(n)); while(!isok(k, n, d), k++); k; \\ Michel Marcus, May 17 2022

Python

def ok(k, n):
    if k == n: return False
    sk, sn = str(k), str(n)
    return all(sk.count(d) >= sn.count(d) for d in set(sn))
def a(n):
    k = 0
    while not ok(k, n): k += 1
    return k
print([a(n) for n in range(71)]) # Michael S. Branicky, May 23 2022