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A007376
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The almost-natural numbers: write n in base 10 and juxtapose digits.
(Formerly M0469)
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133
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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
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OFFSET
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0,3
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COMMENTS
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Also called the Barbier infinite word.
This is an example of a non-morphic sequence.
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
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REFERENCES
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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).
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LINKS
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Putnam Competition No. 48, Problem A2, Math. Mag., 61 (1988), 131-134.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a007376 n = a007376_list !! (n-1)
a007376_list = concatMap (map (read . return) . show) [0..] :: [Int]
(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
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CROSSREFS
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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.
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KEYWORD
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base,easy,nice,nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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