A162501 -id:A162501 - cp's OEIS frontend

A162502 Inverse permutation to A162501.

1, 5, 9, 13, 17, 21, 25, 29, 33, 2, 3, 4, 8, 12, 16, 20, 24, 28, 32, 6, 7, 39, 40, 42, 44, 46, 48, 50, 52, 10, 11, 41, 57, 58, 60, 62, 64, 66, 68, 14, 15, 43, 59, 73, 74, 76, 78, 80, 82, 18, 19, 45, 61, 75, 87, 88, 90, 92, 94, 22, 23, 47, 63, 77, 89, 99, 100, 102, 104, 26, 27

Offset: 1

Reinhard Zumkeller, Jul 05 2009

nonn

A162501(a(n)) = a(A162501(n)) = n.

a(a(n)) = A162504(n).

A162503 A162501(A162501(n)).

Original entry on oeis.org

1, 30, 31, 14, 10, 16, 6, 4, 11, 80, 81, 40, 12, 23, 32, 41, 2, 28, 82, 15, 20, 35, 53, 5, 21, 300, 301, 50, 13, 48, 84, 51, 3, 57, 75, 67, 76, 68, 60, 61, 19, 17, 24, 7, 29, 70, 36, 71, 104, 18, 49, 8, 58, 118, 808, 86, 9, 90, 42, 91, 92, 100, 63, 101, 44, 102, 94, 22, 85, 719

Offset: 1

Reinhard Zumkeller, Jul 05 2009

nonn

Inverse of A162504: a(A162504(n)) = A162504(a(n)) = n;

A162501(n) = A162502(a(n)) = a(A162502(n)).

Index entries for sequences that are permutations of the natural numbers

A184992 a(n) is the least positive integer not occurring earlier that shares a digit with a(n-1); a(1)=1.

Original entry on oeis.org

1, 10, 11, 12, 2, 20, 21, 13, 3, 23, 22, 24, 4, 14, 15, 5, 25, 26, 6, 16, 17, 7, 27, 28, 8, 18, 19, 9, 29, 32, 30, 31, 33, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 44, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 55, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 66, 67, 68, 69, 76, 70, 71, 72, 73

Offset: 1

Eric Angelini, Dec 22 2011

A permutation of the positive integers.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A162501, A076654, A130571.
a(n) = A107353(n) for n>=3. - Alois P. Heinz, Dec 22 2011
Cf. A227118 (inverse); A067581.

import Data.List (delete, intersect); import Data.Function (on)
a184992 n = a184992_list !! (n-1)
a184992_list = 1 : f 1 [2..] where
   f u vs = v : f v (delete v vs)
     where v : _ = filter (not . null . (intersect `on` show) u) vs
-- Reinhard Zumkeller, Jul 01 2013

FromDigits /@ Nest[Function[a, Append[a, Block[{k = 2, d}, While[Nand[FreeQ[a, #], IntersectingQ[a[[-1]], #]] &@ Set[d, IntegerDigits@ k], k++]; d]]], {{1}}, 73] (* Michael De Vlieger, Mar 17 2018 *)

A184992(n,show=0)={my(a=1,u=2^1);for(k=2,n,show && print1(a",");a=Set(Vec(Str(a))); for(j=2,9e9,bittest(u,j) && next;setintersect(Set(Vec(Str(j))),a) || next; u+=2^a=j; break));a}  \\ M. F. Hasler, Dec 22 2011

from itertools import count, islice
def agen(): # generator of terms
    an, aset, mink = 1, {1}, 1
    while True:
        yield an
        digset = set(str(an))
        an = next(k for k in count(mink) if k not in aset and set(str(k))&digset)
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Oct 03 2024

A162504 A162502(A162502(n)).

Original entry on oeis.org

1, 17, 33, 8, 24, 7, 44, 52, 57, 5, 9, 13, 29, 4, 20, 6, 42, 50, 41, 21, 25, 68, 14, 43, 73, 76, 80, 18, 45, 2, 3, 15, 90, 92, 22, 47, 77, 99, 102, 12, 16, 59, 94, 65, 79, 101, 110, 30, 51, 28, 32, 74, 23, 91, 111, 117, 34, 53, 83, 39, 40, 78, 63, 109, 118, 123, 36, 38, 72, 46, 48

Offset: 1

Reinhard Zumkeller, Jul 05 2009

nonn

Inverse of A162503: a(A162503(n)) = A162503(a(n)) = n;

A162502(n) = A162501(a(n)) = a(A162501(n)).

Index entries for sequences that are permutations of the natural numbers

A249494 Lexicographically earliest permutation of the positive integers such that the parity of the first digit of a(n+1) equals that of a(n)'s last digit.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 2, 4, 6, 8, 20, 21, 11, 12, 22, 23, 13, 14, 24, 25, 15, 16, 26, 27, 17, 18, 28, 29, 19, 30, 40, 41, 31, 32, 42, 43, 33, 34, 44, 45, 35, 36, 46, 47, 37, 38, 48, 49, 39, 50, 60, 61, 51, 52, 62, 63, 53, 54, 64, 65, 55, 56, 66, 67, 57, 58, 68, 69, 59, 70, 80, 81, 71, 72, 82, 83, 73, 74, 84, 85, 75, 76, 86, 87, 77, 78, 88, 89, 79, 90, 200

Offset: 1

M. F. Hasler, Oct 30 2014

See A249506 for the inverse permutation.

See A249278 (and inverse A249279) for the variant based on nonnegative integers. A162501 is a variant based not on parity but on equality.

Index entries for sequences that are permutations of the natural numbers

Cf. A249506 (inverse permutation), A249278 (variant starting with 0), A249279 (inverse thereof), A162501 (variant based on equality of digits).

Cf. A000030 (initial digit), A010879 (final digit of n).

a(n,a=1,u=0)=for(n=1,n,for(k=1,9e9, !bittest(u,k)&& k\10^(#Str(k)-1)==Mod(a,2)&& !print1(a=k",")&& break);u+=1<

A249506 Inverse permutation to A249494 (= permutation of the positive integers such that the parity of the first digit of a(n+1) equals that of a(n)'s last digit).

Original entry on oeis.org

1, 7, 2, 8, 3, 9, 4, 10, 5, 6, 13, 14, 17, 18, 21, 22, 25, 26, 29, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 30, 33, 34, 37, 38, 41, 42, 45, 46, 49, 31, 32, 35, 36, 39, 40, 43, 44, 47, 48, 50, 53, 54, 57, 58, 61, 62, 65, 66, 69, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 70, 73, 74, 77, 78, 81, 82, 85

Offset: 1

M. F. Hasler, Oct 30 2014

nonn

See A249278 and A249279 for the variant based on nonnegative integers; A162501 is a variant based not on parity but on equality.

Index entries for sequences that are permutations of the natural numbers

Cf. A000030, A162501.

a=vector(#A249494);for(i=1,#a,A249494[i]<=#a&&a[A249494[i]]=i);a \\ Result only valid up to the first zero.

A319154 a(n) is the smallest nonnegative integer not yet in the sequence that starts with the ending digit of a(n-1); a(1)=0; initial zeros are dropped.

Original entry on oeis.org

0, 1, 10, 2, 20, 3, 30, 4, 40, 5, 50, 6, 60, 7, 70, 8, 80, 9, 90, 11, 12, 21, 13, 31, 14, 41, 15, 51, 16, 61, 17, 71, 18, 81, 19, 91, 100, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 200, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 300, 44, 45, 54

Offset: 1

Enrique Navarrete, Sep 25 2018

Theorem: Every nonnegative number appears.
Proof: (Sketched by Enrique Navarrete, Sep 25 2018; completed by N. J. A. Sloane, Oct 27 2018)
(i) Sequence is infinite (dG, G=giant number, is always available)
(ii) As usual for these "lexicographically earliest distinct term sequences", for any k, there is a threshold n_k such that for all n > n_k, a(n) > k.
(iii) Some final digit (d, say) appears infinitely often. (Otherwise sequence would be finite.) If d=0, go to step (vi).
(iv) All numbers beginning with d appear (If dm were missing, find xd in sequence which is > dm and also > n_{dm}. Then term after xd would be dm, contradiction.)
(v) In particular, all numbers dm0 appear.
(vi) After a number ending in 0, the next number is the smallest missing number. So if x is missing, find dm0 > n_x, then the next term would be (0)x = x, a contradiction. QED

			a(2) = 1 since it is formed from a(1) = 0 as 01 = 1.
a(20) = 11 since it is the smallest number not yet in the sequence that starts with the ending digit 0 of a(19) = 90.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A008592, A162501, A173237, A261370.

Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], If[# == 0, True, First@ IntegerDigits@ k == #] &@ Mod[#[[-1]], 10]], k++]; k]] &, {0}, 69] (* Michael De Vlieger, Oct 15 2018 *)

nexta(v, x) = {my(d = x % 10, newa); for (i=0, oo, newa = eval(concat(Str(d), Str(i))); if (! vecsearch(v, newa), return (newa)););}
lista(nn) = {lasta = 0; print1(lasta, ", "); va = [lasta]; for (n=1, nn, newa = nexta(va, lasta); print1(newa, ", "); va = vecsort(concat(va, newa)); lasta = newa;);} \\ Michel Marcus, Oct 15 2018

A353888 a(n) is the least positive integer not occurring earlier in the sequence that contains at least one digit not in a(n-1); a(1)=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 11, 21, 23, 24, 25, 26, 27, 28, 29, 30, 22, 31, 32, 34, 35, 36, 37, 38, 39, 40, 33, 41, 42, 43, 45, 46, 47, 48, 49, 50, 44, 51, 52, 53, 54, 56, 57, 58, 59, 60, 55, 61, 62, 63, 64, 65, 67, 68

Offset: 1

Sergio Pimentel, May 09 2022

The sequence is finite. 1023456789 should be the last number in the sequence, although many smaller numbers should fail to appear.  How many terms are in the complete sequence?
The last term is a(1023445778) = 1023456789, the least missing number is 1000000010. - Rémy Sigrist, Jun 03 2022
At that point, the least missing numbers containing the digits 2..9 are 1020001000, 1023001300, 1023401000, 1023450200, 1023456024, 1023456710, 1023456781, 1023456789, resp. - Michael S. Branicky, Aug 26 2022

			a(11)=12 since a(10)=10 and 12 is the smallest number not occurring earlier in the sequence that contains a digit (2) that is not in 10.

Sergio Pimentel, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program

Cf. A184992, A162501, A130571, A107353.

isok(k, prev) = {my(d=digits(k)); for (i=1, #d, if (!vecsearch(prev, d[i]), return(1));); return(0);}
find(va, n) = {my(k=1, prev=Set(digits(va[n-1]))); while (vecsearch(Set(va), k) || !isok(k, prev), k++); k;}
lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = find(va, n);); va;} \\ Michel Marcus, May 11 2022
(C++) See Links section.

from itertools import count, islice
def agen():  # generator of terms
    an, aset, mu, mink = 0, set(), [10, 1, 2, 3, 4, 5, 6, 7, 8, 9], 1
    while set(str(an)) != set("0123456789"):
        notin = set("0123456789") - set(str(an))
        an = min(mu[i] for i in range(10) if str(i) in notin)
        yield an; aset.add(an)
        for i in range(10):  # update min unused containing digit i
            while mu[i] in aset or str(i) not in str(mu[i]): mu[i] += 1
        for k in range(mink, min(mu)): aset.discard(k)
        mink = min(mu)
print(list(islice(agen(), 67))) # Michael S. Branicky, Aug 26 2022

cp's OEIS Frontend

A162502 Inverse permutation to A162501.

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A162503 A162501(A162501(n)).

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A184992 a(n) is the least positive integer not occurring earlier that shares a digit with a(n-1); a(1)=1.

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A162504 A162502(A162502(n)).

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A249494 Lexicographically earliest permutation of the positive integers such that the parity of the first digit of a(n+1) equals that of a(n)'s last digit.

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A249506 Inverse permutation to A249494 (= permutation of the positive integers such that the parity of the first digit of a(n+1) equals that of a(n)'s last digit).

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A319154 a(n) is the smallest nonnegative integer not yet in the sequence that starts with the ending digit of a(n-1); a(1)=0; initial zeros are dropped.

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A353888 a(n) is the least positive integer not occurring earlier in the sequence that contains at least one digit not in a(n-1); a(1)=1.

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