A358497 Replace each new digit in n with index 1, 2, ..., 9, 0 in order in which that digit appears in n, from left to right.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 12

Offset: 0

Author

Gleb Ivanov, Nov 19 2022

Keywords

• nonn
• base
• easy

Comments

a(n) gives the canonical form which enumerates digits in order of their first occurrence in n, from left to right. a(n) uniquely defines the pattern of identical digits in n. - Dmytro Inosov, Jul 16 2024

Examples

			n = 10 has 2 different digits; replace first encountered digit 1 -> 1, replace second digit 0 -> 2, therefore a(10) = 12.
n = 141 has 3 digits, but only 2 different ones; replace first encountered digit 1 -> 1, replace second encountered digit 4 -> 2, therefore a(141) = 121.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999
• Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3, 18.

Crossrefs

Cf. A071159, A227362, A358615 (record high values).

Programs

Mathematica

A358497[k_] := With[{pI = Values@PositionIndex@IntegerDigits@k}, MapIndexed[#1 -> Mod[#2[[1]], 10] &, pI, {2}] // Flatten // SparseArray // FromDigits]; (* Dmytro Inosov, Jul 15 2024 *)

PARI

a(n) = {if(n == 0, return(1)); my(d = digits(n), m = Map(), t = 0); for(i = 1, #d, if(mapisdefined(m, d[i]), d[i] = mapget(m, d[i]) , t++; if(t == 10, t = 0); mapput(m, d[i], t); d[i] = t ) ); fromdigits(d) } \\ David A. Corneth, Nov 23 2022

Python

def A358497(n):
    d,s,k = dict(),str(n),1
    for i in range(len(s)):
        if d.get(s[i],0) == 0:
            d[s[i]] = str(k)
            k = (k + 1)%10
    s_t = list(s)
    for i in range(len(s)):s_t[i] = d[s[i]]
    return int(''.join(s_t))
print([A358497(i) for i in range(100)])

Python

def A358497(n):
    s, c, i  = str(n), {}, 49
    for d in map(ord,s):
        if d not in c:
            c[d] = i
            i += 1
    return int(s.translate(c)) # Chai Wah Wu, Jul 09 2024

Formula

a(a(n)) = a(n). - Dmytro Inosov, Jul 16 2024