A007376 -id:A007376 - cp's OEIS frontend

A031297 a(n) is the least k such that the base-10 representation of n begins at s(k), where s=A007376.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, 16, 18, 20, 22, 24, 26, 28, 30, 15, 34, 2, 38, 40, 42, 44, 46, 48, 50, 17, 37, 56, 3, 60, 62, 64, 66, 68, 70, 19, 39, 59, 78, 4, 82, 84, 86, 88, 90, 21, 41, 61, 81, 100, 5, 104, 106, 108, 110, 23, 43, 63, 83

Offset: 1

Clark Kimberling

A229186 is the same sequence including the a(0) term.

			a(1) = a(12) = a(123) = 1 since they each start at index 1 in 0123.
a(21) = 15 since it appears first at index 15 in 012345678910111213.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A007376, A229186 (same sequence but including the a(0) term).
Cf. A165449.
Cf. A030304 (binary variant).

with(StringTools): s:="": for n from 1 to 70 do s:=cat(s,convert(n,string)): od: seq(Search(convert(n, string), s), n=1..70); # Nathaniel Johnston, May 26 2011

nmax = 100;
s = Table[IntegerDigits[n], {n, 0, nmax}] // Flatten;
a[n_] := SequencePosition[s, IntegerDigits[n], 1][[1, 1]] - 1;
Array[a, nmax] (* Jean-François Alcover, Feb 21 2021 *)

from itertools import count, islice
def agen():
    k, chap = 0, "0"
    for n in count(1):
        target = str(n)
        while chap.find(target) == -1: k += 1; chap += str(k)
        yield chap.find(target)
print(list(islice(agen(), 70))) # Michael S. Branicky, Oct 06 2022

A098067 Consider the succession of single digits of the positive integers: 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 ... (A007376). This sequence is the lexicographically earliest derangement of the positive integers that produces the same succession of digits.

Original entry on oeis.org

12, 3, 4, 5, 6, 7, 8, 9, 10, 1, 112, 13, 14, 15, 16, 17, 18, 19, 20, 2, 122, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Eric Angelini, Sep 13 2004

Derangement here means that a(n) != n.

			We must begin with "1,2,3,..." and we cannot have a(1) = 1, so the first possible term is "12". The next term must be the smallest available positive integer not leading to a contradiction, thus "3"; the next one will be "4"; etc. After a(10) = 1, we cannot have a(11) = 11, so we use "112" instead. We are not allowed to use "2" after "19" because the next term would have a leading zero, which is forbidden. - _Eric Angelini_, Aug 12 2008

Paul Tek, Table of n, a(n) for n = 1..10000
Eric Angelini, Jeux de suites, in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
Paul Tek, PERL program for this sequence

Cf. A097487, A097968, A098099, A228595.

lim = 80; f[lst_List, k_] := Block[{L = lst, g, a = {}, m = 0}, g[] := {Set[m, First@ FromDigits@ Append[IntegerDigits@ m, First@ #]], Set[L, Last@ #]} &@ TakeDrop[L, 1]; Do[g[]; While[Or[m == Length@ a + 1, First@ L == 0, MemberQ[a, m]], g[]]; AppendTo[a, m]; m = 0, {k}]; a]; f[Flatten@ Map[IntegerDigits, Range@ lim], Floor[lim - 10^(Log10@ lim - 1)]] (* Michael De Vlieger, Nov 29 2015, Version 10.2 *)

Perl
```
See Link section.
```

from itertools import count
def diggen():
    for k in count(1): yield from list(map(int, str(k)))
def aupton(terms):
    g = diggen()
    alst, aset, , , nextd = [12], {12}, next(g), next(g), next(g)
    for n in range(2, terms+1):
        an, nextd = nextd, next(g)
        while an in aset or an == n or nextd == 0:
            an, nextd = int(str(an) + str(nextd)), next(g)
        alst.append(an); aset.add(an)
    return alst
print(aupton(72)) # Michael S. Branicky, Dec 03 2021

Corrected and extended by Jacques ALARDET and Eric Angelini, Aug 12 2008
Derangement wording introduced by Danny Rorabaugh, Nov 26 2015
Edited by Danny Rorabaugh, Nov 29 2015

A256100 In S = A007376 (read as a sequence) the digit S(n) appears a(n) times in the sequence S(1), ..., S(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 4, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 3, 2, 4, 13, 5, 6, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 4, 3, 5, 14, 6, 14, 7, 8, 9, 4, 10, 4, 11, 4, 12, 4, 13, 4, 14, 4, 5, 4, 6, 15, 7, 15, 8, 15, 9, 10, 11, 5, 12, 5, 13, 5, 14, 5, 15, 5, 6

Offset: 1

Wolfdieter Lang, Apr 08 2015

The motivation to consider this sequence came from the proposal A256379 by Anthony Sand.
This sequence can also be read as an irregular triangle (array) in which a(n, k) is the number of appearances of the k-th digit of n in the digits of 1, ... ,n-1 and the first k digits of n. See the example for the head of this array. The row length is A055842(n), n >= 1.
This can also be described as the ordinal transform of A007376. - Franklin T. Adams-Watters, Oct 10 2015

			a(10) = 2 because A007376(10) = 1 and that sequence up to n=10 is 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, and 1 appears twice.
a(24) = 10 because A007376(24) = 1 and this is the tenth 1 in A007376 up to, and including, A007376(24).
Read as a tabf array a(n, k) with row length A055842(n) this begins:
   n\k  1  2  ...
   1:   1
   2:   1
   3:   1
   4:   1
   5:   1
   6:   1
   7:   1
   8:   1
   9:   1
  10:   2  1
  11:   3  4
  12:   5  2
  13:   6  2
  14:   7  2
  15:   8  2
  16:   9  2
  17:  10  2
  18:  11  2
  19:  12  2
  20:   3  2
  ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007376, A065648.

a256100 n = a256100_list !! (n-1)
a256100_list = f a007376_list $ take 10 $ repeat 1 where
   f (d:ds) counts = y : f ds (xs ++ (y + 1) : ys) where
                           (xs, y:ys) = splitAt d counts
-- Reinhard Zumkeller, Aug 13 2015

lim = 120; s = Flatten[IntegerDigits /@ Range@ lim]; f[n_] := Block[{d = IntegerDigits /@ Take[s, n] // Flatten // FromDigits}, DigitCount[d][[If[ s[[n]] == 0, 10, s[[n]] ]] ] ]; Array[f, lim] (* Michael De Vlieger, Apr 08 2015, after Robert G. Wilson v at A007376 *)

a(n) gives the number of digits A007376(n) in the sequence starting with A007376(1) and ending with A007376(n).

A054632 Partial sums of A007376.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 46, 47, 48, 49, 51, 52, 55, 56, 60, 61, 66, 67, 73, 74, 81, 82, 90, 91, 100, 102, 102, 104, 105, 107, 109, 111, 114, 116, 120, 122, 127, 129, 135, 137, 144, 146, 154, 156, 165, 168, 168, 171, 172, 175, 177

Offset: 0

N. J. A. Sloane, Apr 16 2000

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

a054632 n = a054632_list !! n
a054632_list = scanl (+) 0 a007376_list
-- Reinhard Zumkeller, Dec 29 2011

from itertools import accumulate, count, islice
def A054632gen(): return accumulate(int(d) for n in count(0) for d in str(n))
A054632_list = list(islice(A054632gen(),40)) # Chai Wah Wu, Dec 04 2021

A108202 a(2n) = A007376(n); a(2n+1) = a(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 4, 2, 5, 1, 6, 3, 7, 0, 8, 4, 9, 2, 1, 5, 0, 1, 1, 6, 1, 3, 1, 7, 2, 0, 1, 8, 3, 4, 1, 9, 4, 2, 1, 1, 5, 5, 1, 0, 6, 1, 1, 1, 7, 6, 1, 1, 8, 3, 1, 1, 9, 7, 2, 2, 0, 0, 2, 1, 1, 8, 2, 3, 2, 4, 2, 1, 3, 9, 2, 4, 4, 2, 2, 1, 5, 1, 2, 5, 6, 5, 2, 1, 7, 0, 2, 6, 8, 1, 2, 1, 9, 1, 3, 7, 0, 6, 3

Offset: 0

Juliette Bruyndonckx and Alexandre Wajnberg, Jun 15 2005

A sequence, S, consisting of the natural counting digits (A007376) interleaved with S.
Fractal sequence (Kimberling-type) based upon the counting digits.
Similar to but different from A025480 - N. J. A. Sloane, Nov 26 2020.

First differs from A025480 at a(20).
Even bisection: A007376.
Cf. A003602.

a(n) = A007376(A025480(n)). - Kevin Ryde, Nov 21 2020

More terms from Joshua Zucker, May 18 2006

Name made more explicit by Peter Munn, Nov 20 2020

A098727 Consider the sequence {b(n), n >= 1} of digits of the natural (or counting) numbers: 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0... (A007376); a(n) = b(n) + n.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 11, 13, 14, 15, 17, 17, 20, 19, 23, 21, 26, 23, 29, 25, 32, 27, 35, 29, 38, 32, 31, 34, 34, 36, 37, 38, 40, 40, 43, 42, 46, 44, 49, 46, 52, 48, 55, 50, 58, 53, 51, 55, 54, 57, 57, 59, 60, 61, 63, 63, 66, 65, 69, 67, 72, 69, 75, 71, 78, 74, 71

Offset: 1

Alexandre Wajnberg, Sep 30 2004

Add each digit of the counting numbers to its rank.

			The sequence of digits of the counting numbers is
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0...
The 15th term, for instance, is a 2. Thus 2+15=17 is the 15th term of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007376, A033307, A098728, A098729, A098732, A098733, A098734.

Module[{dcn=Flatten[IntegerDigits/@Range[70]]},Total/@Thread[ {dcn,Range[ Length[dcn]]}]] (* Harvey P. Dale, Mar 25 2022 *)

More terms from Joshua Zucker, May 18 2006

A127050 A007376(2n+1).

Original entry on oeis.org

1, 3, 5, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 0, 1, 3, 0, 1, 5, 0

Offset: 0

N. J. A. Sloane, Jul 23 2008

This and the other "decimations" of A007376 were suggested by a problem in Honsberger's book.

R. Honsberger, Mathematical Chestnuts from Around the World, MAA, 2001; see p. 163.

List of decimations of A007376: A127050 A127353 A127414 A127508 A127584 A127734 A127794 A127950 A128178 A128211 A128359 A128423 A128475 A128881.

from itertools import islice, count
def A127050gen(): return islice((int(d) for n in count(0) for d in str(n)),1,None,2)
A127050_list = list(islice(A127050gen(),40)) # Chai Wah Wu, Dec 04 2021

A127353 A007376(2n).

Original entry on oeis.org

2, 4, 6, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 1, 0, 0, 1, 2, 0, 1, 4, 0, 1, 6

Offset: 1

N. J. A. Sloane, Jul 23 2008

from itertools import islice, count
def A127353gen(): return islice((int(d) for n in count(0) for d in str(n)),2,None,2)
A127533_list = list(islice(A127353gen(),30)) # Chai Wah Wu, Dec 04 2021

A261333 a(n) = 10*(A256100(n+1)-1) + A007376(n+1), a simple variation of A065649.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 0, 21, 31, 41, 12, 51, 13, 61, 14, 71, 15, 81, 16, 91, 17, 101, 18, 111, 19, 22, 10, 32, 121, 42, 52, 62, 23, 72, 24, 82, 25, 92, 26, 102, 27, 112, 28, 122, 29, 33, 20, 43, 131, 53, 132, 63, 73, 83, 34, 93, 35, 103, 36, 113, 37

Offset: 0

Reinhard Zumkeller, Aug 15 2015

Permutation of the nonnegatve integers with inverse A261334.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A256100, A007376, A261334 (inverse), A261335 (fixed points), A065649.

a261333 n = a261333_list !! n
a261333_list = zipWith (+)
               (map ((* 10) . subtract 1) a256100_list) a007376_list

A055059 Positions in A007376 where the partial sums (A054632) yield primes (A055058).

Original entry on oeis.org

3, 13, 21, 23, 24, 35, 36, 42, 45, 61, 83, 85, 114, 115, 135, 141, 149, 181, 200, 201, 203, 204, 212, 213, 215, 216, 234, 237, 239, 242, 247, 251, 260, 263, 272, 285, 292, 297, 309, 311, 332, 335, 347, 350, 357, 359, 371, 392, 396, 405, 410, 417, 419, 431

Offset: 1

Patrick De Geest, Apr 15 2000

			0123456789101112131415161718192021...
--|.................. sum 0+1+2 = prime 3 at position 3
------------|........ sum 0+1+2+3+4+5+6+7+8+9+1+0+1 = prime 47 at position 13
--------------------| sum 0+1+2+3+4+5+6+7+8+9+1+0+1+1+1+2+1+3+1+4+1 = prime 61 at position 21

Cf. A007376, A054632, A055058.

Definition amended and offset changed by Georg Fischer, Sep 03 2022

cp's OEIS Frontend

A031297 a(n) is the least k such that the base-10 representation of n begins at s(k), where s=A007376.

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A098067 Consider the succession of single digits of the positive integers: 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 ... (A007376). This sequence is the lexicographically earliest derangement of the positive integers that produces the same succession of digits.

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A256100 In S = A007376 (read as a sequence) the digit S(n) appears a(n) times in the sequence S(1), ..., S(n).

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A054632 Partial sums of A007376.

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A108202 a(2n) = A007376(n); a(2n+1) = a(n).

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A098727 Consider the sequence {b(n), n >= 1} of digits of the natural (or counting) numbers: 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0... (A007376); a(n) = b(n) + n.

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A127050 A007376(2n+1).

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A127353 A007376(2n).

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