A269631 a(1) = 0; a(n+1) is the smallest integer not yet used that contains the number of decimal digits of a(n) as a substring.

0, 1, 10, 2, 11, 12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92, 102, 3, 13, 112, 30, 120, 31, 121, 33, 122, 34, 123, 35, 124, 36, 125, 37, 126, 38, 127, 39, 128, 43, 129, 53, 132, 63, 142, 73, 152, 83, 162, 93, 172, 103, 113, 130, 131, 133, 134, 135

Offset

1,3

Comments

Conjecture: a(n) is a permutation of the nonnegative integers.

The following table shows:

C = number of terms calculated

F = number of terms less than C

+------------+----------+-----------+

| C | F | % |

+------------+----------+------------

| 10 | 2 | 20.0 |

| 100 | 39 | 39.0 |

| 1000 | 532 | 53.2 |

| 10000 | 6379 | 63.79 |

| 100000 | 71609 | 71.609 |

| 1000000 | 765630 | 76.563 |

| 10000000 | 7907944 | 79.07944 |

| 100000000 | 81251152 | 81.251152 |

.

and the growth of percentage seems to support the conjecture.

Links

Francesco Di Matteo, Table of n, a(n) for n = 1..1000

Example

a(2) = 1 because a(1) = 0 and 0 has 1 decimal digit;
a(3) = 10 because a(2) = 1, so 1 has 1 digit, and 10 is the first integer not yet used that contains "1";
a(4) = 2 because a(3) = 10 and 10 has 2 digits;
a(5) = 11 because a(4) = 2, so 2 has 1 digit, and 11 is the first integer not yet used that contains "1"; ...

Mathematica

a = {0, 1}; Do[AppendTo[a, SelectFirst[Range[10^3], And[! MemberQ[a, #], MemberQ[IntegerDigits@ #, IntegerLength@ a[[n - 1]]]] &]], {n, 3, 64}]; a (* Michael De Vlieger, Apr 01 2016 *)

Prog

(Python)
# This routine is a little bit more complex compared to the same with
# the a(n) terms 'storage' (now we memorize only the highest number
# for every k-digits) but is very much faster.
print("0", end=', ')
lista = [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
a = 1
for g in range (1, 100):
    b = len(str(a))
    val = lista[b] + 1
    flag = 0
    while flag == 0:
        sval = str(val)
        while str(b) not in sval:
            val += 1
            sval = str(val)
        k = len(sval)
        comp = k
        for s in range(k):
            if lista[int(sval[s])] < val:
                comp -= 1
        if comp == 0:
            flag = 1
        else:
            val +=1
    print(val, end=', ')
    lista[b] = val
    a = val
# Francesco Di Matteo, Mar 02 2016
(PARI) findnew(nbd, vsa) = {k=0; while (vecsearch(vsa, k) || !vecsearch(vecsort(digits(k)), nbd), k++); k; }
listd(nn) = {va = vector(nn); print1(va[1], ", "); vsa = vecsort(va, , 8); for (n=2, nn, nbd = #Str(va[n-1]); na = findnew(nbd, vsa); print1(na, ", "); va[n] = na; vsa = vecsort(va, , 8); ); } \\ Michel Marcus, Mar 08 2016

Crossrefs

Sequence in context: A351840 A257277 A248024 * A334737 A334837 A327689

Adjacent sequences: A269628 A269629 A269630 * A269632 A269633 A269634

Keyword

nonn,base

Author

Francesco Di Matteo, Mar 01 2016

Status

approved