A248024 -id:A248024 - cp's OEIS frontend

A248043 Inverse permutation to A248024.

1, 3, 6, 13, 11, 16, 9, 27, 32, 2, 4, 5, 7, 8, 10, 12, 14, 15, 17, 19, 20, 22, 23, 26, 28, 30, 31, 36, 37, 21, 33, 39, 45, 46, 50, 53, 54, 56, 61, 40, 41, 52, 58, 65, 66, 69, 74, 80, 81, 29, 49, 51, 63, 67, 72, 73, 78, 88, 94, 68, 90, 99, 102, 113, 122, 130

Offset: 1

Reinhard Zumkeller, Sep 30 2014

nonn

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

import Data.List (elemIndex); import Data.Maybe (fromJust)
a248043 = (+ 1) . fromJust . (`elemIndex` a248024_list)

A248025 Lexicographically earliest permutation of the positive integers such that the first digit of a(n+1) is the digital root of a(n).

Original entry on oeis.org

1, 10, 11, 2, 20, 21, 3, 30, 31, 4, 40, 41, 5, 50, 51, 6, 60, 61, 7, 70, 71, 8, 80, 81, 9, 90, 91, 12, 32, 52, 72, 92, 22, 42, 62, 82, 13, 43, 73, 14, 53, 83, 23, 54, 93, 33, 63, 94, 44, 84, 34, 74, 24, 64, 15, 65, 25, 75, 35, 85, 45, 95, 55, 16, 76, 46, 17, 86, 56, 26, 87, 66, 36, 96

Offset: 1

Eric Angelini and M. F. Hasler, Sep 29 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Angelini, Fun and quick permutation (with a digital root), SeqFan list, Sep 29 2014.
Index entries for sequences that are permutations of the natural numbers

Cf. A010888 (digital root); similar sequences: A248024,...

Cf. A247879 (inverse), A000030.

import Data.List (delete)
a248025 n = a248025_list !! (n-1)
a248025_list = 1 : f 1 [2..] where
  f x zs = g zs where
    g (y:ys) = if a000030 y == a010888 x
               then y : f y (delete y zs) else g ys
-- Reinhard Zumkeller, Sep 30 2014

Nest[Append[#, Block[{k = 1, r = Mod[#[[-1]], 9] + 9 Boole[Mod[#[[-1]], 9] == 0]}, While[Nand[FreeQ[#, k], IntegerDigits[k][[1]] == r], k++]; k]] &, {1}, 73] (* Michael De Vlieger, Oct 15 2020 *)

a(n,S=1,u=2)={for(i=1,n,print1(S",");S=(S-1)%9+1;for(k=1,9e9,bittest(u,k)&&next;S==digits(k)[1]||next;u+=1<

A308539 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the initial digit of a(n) divides a(n+1).

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 6, 12, 5, 10, 7, 14, 9, 18, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 21, 32, 27, 34, 33, 36, 39, 42, 40, 44, 48, 52, 25, 38, 45, 56, 35, 51, 50, 55, 60, 54, 65, 66, 72, 49, 64, 78, 63, 84, 80, 88, 96, 81, 104, 23, 46, 68, 90, 99, 108, 29

Offset: 1

Rémy Sigrist, Jun 06 2019

This sequence is a permutation of the natural numbers (with inverse A308541):
- for any nonzero digit d, there are infinitely many multiples of d, hence we can always extend the sequence,
- by the pigeonhole principle, for some nonzero digit t, there are infinitely many terms with initial digit t,
- so eventually every multiple of t will appear in the sequence,
- after a term with initial digit 1, we can always extend the sequence with the least natural number not yet in the sequence,
- as there are infinitely many multiples of t with initial digit 1, so infinitely many terms with initial digit 1, every natural number will eventually appear, QED.

			a(1) = 1.
a(2) = 2 as it is the first multiple of 1 not yet in the sequence.
a(3) = 4 as it is the first multiple of 2 not yet in the sequence.
a(4) = 8 as it is the first multiple of 4 not yet in the sequence.
a(5) = 16 as it is the first multiple of 8 not yet in the sequence.
a(6) = 3 as it is the first multiple of 1 not yet in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of the initial digits of a(n-1))
Index entries for sequences that are permutations of the natural numbers

See A248024 for a similar sequence.

Cf. A000030, A308541 (inverse).

a[1] = 1; a[n_] := a[n] = Block[{k = 2}, While[Mod[k, First@IntegerDigits[a[n - 1]]] != 0 || MemberQ[Array[a, n - 1], k], k++]; k]; Array[a, 67] (* Giorgos Kalogeropoulos, May 12 2023 *)

{ s=0; v=1; u=1; for (n=1, 67, print1 (v ", "); s+=2^v; while (bittest(s,u), u++); forstep (w=ceil(u/d=digits(v)[1])*d, oo, d, if (!bittest(s,w), v=w; break))) }

from itertools import count, islice
def agen(): # generator of terms
    an, aset, m = 1, {1}, 1
    while True:
        yield an
        an1 = int(str(an)[0])
        an = next(k for k in count(m) if k not in aset and k%an1 == 0)
        aset.add(an)
        while m in aset: m += 1
print(list(islice(agen(), 67))) # Michael S. Branicky, Jan 27 2025

A257277 a(n+1) has a digit that divides a(n) and is the least positive integer not appearing earlier with this property.

Original entry on oeis.org

0, 1, 10, 2, 11, 12, 3, 13, 14, 7, 15, 5, 16, 4, 17, 18, 6, 19, 21, 23, 31, 41, 51, 30, 20, 22, 24, 8, 25, 35, 27, 9, 29, 61, 71, 81, 32, 26, 28, 34, 42, 33, 36, 37, 91, 47, 100, 40, 38, 52, 43, 101, 102, 39, 53, 103, 104, 44, 45, 49, 57, 63, 59, 105, 50, 54, 46, 62, 72, 48, 56, 58, 82, 92, 64, 68, 74, 106, 107, 108, 60, 55, 65, 75, 73, 109, 110, 85

Offset: 0

Eric Angelini and M. F. Hasler, May 07 2015

A variant of A248024.
This is a permutation of the nonnegative integers, see A257276 for the inverse permutation.
There are large ranges of fixed points, e.g., between a(135) = 99 and a(200) = 201, or between a(1080) = 999 and a(2000) = 2001. For indices n in these ranges, the sequence restricted to [0...n] is a permutation (i.e., all numbers up to n appear among the values up to that point).
If one requires that a(n+1) has *no* digit dividing a(n), the sequence starts 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 30, 40, 33, 22, 34, ... and stops at a(1422) = 2520, divisible by any digit. If one requires the sequence to be increasing, then it goes ..., 20, 30, 40, 60, 70, 80, 90, 400, 600, 700, 800, 900, 7000, 9000, 70000, 90000, 700000, 900000, ...

M. F. Hasler, Table of n, a(n) for n = 0..1199
E. Angelini, Division by digits (variation), SeqFan list, May 7, 2015.

Cf. A248024, A257276.

{u=n=0;until(print1(n","),u+=1<!(n%i),vector(9,i,i));n=0;until(!bittest(u,n++)&&setintersect(d,Set(digits(n))),))}

A381081 Lexicographically earliest sequence of distinct positive integers such that the string value of a(n) begins with a divisor of a(n-1).

Original entry on oeis.org

1, 10, 2, 11, 12, 3, 13, 14, 7, 15, 5, 16, 4, 17, 18, 6, 19, 100, 20, 21, 30, 22, 23, 101, 102, 24, 8, 25, 50, 26, 27, 9, 31, 103, 104, 28, 29, 105, 32, 40, 41, 106, 53, 107, 108, 33, 34, 109, 110, 51, 35, 52, 42, 36, 37, 111, 38, 112, 43, 113, 114, 39, 115, 54, 60, 44, 45, 55, 56, 46, 116, 47, 117, 90, 57, 118, 59, 119, 70, 58, 120, 48, 49, 71, 121, 122

Offset: 1

Scott R. Shannon, Feb 13 2025

The sequence contains many fixed points, these beginning 1, 22, 23, 40, 41, 52, 199, 1936, 1937, 1938, 1939, ... .

			a(2) = 10 as the only divisor of a(1) = 1 is 1, and 10 is the smallest unused number to begin with 1.
a(43) = 53 as a(42) = 106 whose divisors are 1, 2, 53, 106, and 53 is the smallest unused number to begin with 53 - all other smaller numbers beginning with 1 and 2 have been used. This is the first term to differ from A248024.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 150000 terms. The green line is a(n) = n.

Cf. A027750, A248024, A000030.

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A248043 Inverse permutation to A248024.

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A248025 Lexicographically earliest permutation of the positive integers such that the first digit of a(n+1) is the digital root of a(n).

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A308539 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the initial digit of a(n) divides a(n+1).

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A257277 a(n+1) has a digit that divides a(n) and is the least positive integer not appearing earlier with this property.

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A381081 Lexicographically earliest sequence of distinct positive integers such that the string value of a(n) begins with a divisor of a(n-1).

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