A007376 The almost-natural numbers: write n in base 10 and juxtapose digits.
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
Offset: 0
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).
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.
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.
- Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993
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.
For "decimations" see A127050 A127353 A127414 A127508 A127584 A127734 A127794 A127950 A128178 A128211 A128359 A128423 A128475 A128881.
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.
Programs
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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
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Magma
&cat[Reverse(IntegerToSequence(n)):n in[0..31]]; // Jason Kimberley, Dec 07 2012
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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
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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 *) -
PARI
for(n=0,90,v=digits(n);for(i=1,#v,print1(v[i]", "))) \\ Charles R Greathouse IV, Nov 20 2012
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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
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Python
A007376_list = [int(d) for n in range(10**2) for d in str(n)] # Chai Wah Wu, Feb 04 2015
Extensions
Extended to a(0) = 0 by M. F. Hasler, Oct 23 2019
Comments