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A007376 The almost-natural numbers: write n in base 10 and juxtapose digits.
(Formerly M0469)
127
0, 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, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 5, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also called the Barbier infinite word.

This is an example of a non-morphic sequence.

a(n) = A162711(n,1); A136414(n) = 10*a(n) + a(n+1). - Reinhard Zumkeller, Jul 11 2009

a(A031287(n)) = 0, a(A031288(n)) = 1, a(A031289(n)) = 2, a(A031290(n)) = 3, a(A031291(n)) = 4, a(A031292(n)) = 5, a(A031293(n)) = 6, a(A031294(n)) = 7, a(A031295(n)) = 8, a(A031296(n)) = 9. - Reinhard Zumkeller, Jul 28 2011

May be regarded as an irregular table in which the n-th row lists the digits of n. - Jason Kimberley, Dec 07 2012

The digits of the integer n start at index A117804(n). The digit a(n) at index n belongs to the number A100470(n). - M. F. Hasler, Oct 23 2019

See also the Copeland-Erdős constant A033308, equivalent using primes instead of all numbers. - M. F. Hasler, Oct 24 2019

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 114, 336.

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

M. Kraitchik, Mathematical Recreations. Dover, NY, 2nd ed., 1953, p. 49.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..100000 (a(0) = 0 added by M. F. Hasler, Oct 23 2019).

Putnam Competition No. 48, Problem A2, Math. Mag., 61 (1988), 131-134.

R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

MAPLE

c:=proc(x, y) local s: s:=proc(m) nops(convert(m, base, 10)) end: if y=0 then 10*x else x*10^s(y)+y: fi end: b:=proc(n) local nn: nn:=convert(n, base, 10):[seq(nn[nops(nn)+1-i], i=1..nops(nn))] end: A:=0: for n from 1 to 75 do A:=c(A, n) od: b(A); # c concatenates 2 numbers while b converts a number to the sequence of its digits - Emeric Deutsch, Jul 27 2006

#alternative

A007376 := proc(n) option remember ; local aprev, dOld, N ; if n <=9 then RETURN([n, n, 1]) ; else aprev := A007376(n-1) ; dOld := op(3, aprev) ; N := op(2, aprev) ; if dOld < A055642(N) then RETURN([op(-dOld-1, convert(N, base, 10)), N, dOld+1]) ; else RETURN([op(-1, convert(N+1, base, 10)), N+1, 1]) ; fi ; fi ; end: # R. J. Mathar, Jan 21 2008

MATHEMATICA

Flatten[ IntegerDigits /@ Range@ 57] (* Or *)

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 10] &, 105] (* updated Jun 29 2014 *)

With[{nn=120}, RealDigits[N[ChampernowneNumber[], nn], 10, nn]][[1]] (* Harvey P. Dale, Mar 13 2018 *)

PROG

(Haskell)

a007376 n = a007376_list !! (n-1)

a007376_list = concatMap (map (read . return) . show) [0..] :: [Int]

-- Reinhard Zumkeller, Nov 11 2013, Dec 17 2011, Mar 28 2011

(PARI) for(n=0, 90, v=digits(n); for(i=1, #v, print1(v[i]", "))) \\ Charles R Greathouse IV, Nov 20 2012

(PARI) apply( A007376(n)={for(k=1, n, k*10^k>n&& return(digits(n\k)[n%k+1]); n+=10^k)}, [0..200]) \\ M. F. Hasler, Nov 03 2019

(MAGMA) &cat[Reverse(IntegerToSequence(n)):n in[0..31]]; // Jason Kimberley, Dec 07 2012

(Python) A007376_list = [int(d) for n in range(10**2) for d in str(n)] # Chai Wah Wu, Feb 04 2015

CROSSREFS

Considered as a sequence of digits, this is the same as the decimal expansion of the Champernowne constant, A033307. See that entry for a formula for a(n), further references, etc.

Cf. A054632 (partial sums), A023103.

For "decimations" see A127050 A127353 A127414 A127508 A127584 A127734 A127794 A127950 A128178 A128211 A128359 A128423 A128475 A128881.

Cf. A193428, A256100, A001477 (the nonnegative integers), A117804, A100470.

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), this sequence and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Row lengths in A055642.

For primes here see A071620. See A007908 for a very similar sequence.

Sequence in context: A048379 A333666 A033307 * A189823 A325479 A001073

Adjacent sequences:  A007373 A007374 A007375 * A007377 A007378 A007379

KEYWORD

base,easy,nice,nonn,tabf

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

Extended to a(0) = 0 by M. F. Hasler, Oct 23 2019

STATUS

approved

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Last modified November 27 23:47 EST 2020. Contains 338685 sequences. (Running on oeis4.)