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Search: id:A007908
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| A007908 |
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Concatenation of the numbers from 1 to n. |
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+0
84
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| 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, 12345678910111213, 1234567891011121314, 123456789101112131415, 12345678910111213141516, 1234567891011121314151617
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Sometimes called Smarandache consecutive numbers.
Also called the triangle of the gods (see Pickover link).
As n -> infinity, lim((A007908(n))/(prod(i=1,n, 10^floor(1+(log(i)/(log(10))))))) yields the Champernowne constant. - Alexander R. Povolotsky (pevnev(AT)juno.com), May 29 2008, Paolo Lava, Jun 06 2008
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REFERENCES
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Y. Guo and M. Le, Smarandache Concatenated Power Decimals and Their Irrationality, Smarandache Notions Journal, Vol. 9, No. 1-2. 1998, 100-102.
R. K. Guy, Unsolved Problems in Number Theory, A3.
F. Smarandache, "Only Problems, not Solutions!", Xiquan Publ., Phoenix-Chicago, 1993.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..100
M. L. Perez et al., eds., Smarandache Notions Journal
Clifford Pickover, Triangle of the Gods
R. W. Stephan, Factors and primes in two Smarandache sequences
F. Smarandache, Only Problems, Not Solutions!
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n)=a(n-1)*10^floor[log10(10*n)]+n - Paolo P. Lava (ppl(AT)spl.at), Feb 01 2008
a(n)=n+a(n-1)*10^A055642(n) - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 31 2008
a(n) = prod(a,1,n,10^floor(1+log(10)^(-1)*log(a))) *sum(b,1,n,prod(a,1,b,10^floor(log(10)^(-1)*(log(10)+log(a))))^(-1)*b). - Alexander R. Povolotsky and Paolo Lava, Jun 06 2008
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MAPLE
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A055642 := proc(n) max(1, ilog10(n)+1) ; end: A007908 := proc(n) if n = 1 then 1; else A007908(n-1)*10^A055642(n)+n ; fi ; end: seq(A007908(n), n=1..12) ; # R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 31 2008
P:=proc(i) local a, b, n, x; for n from 1 by 1 to i do x:=evalf(product(10^floor(1+log10(a)), a=1..n)*sum('product(10^floor(log10(10)+log10(a)), a= 1..b)^(-1)*b', 'b'=1..n)); od; end: - Alexander R. Povolotsky and Paolo Lava, Jun 06 2008
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PROGRAM
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(PARI) A007908(n)= prod(a=1, n, 10^floor(1+log(10)^(-1)*log(a)))*sum(b=1, n, prod(a=1, b, 10^floor(log(10)^(-1)*(log(10)+log(a))))^(-1)*b) - Alexander R. Povolotsky and Paolo Lava, Jun 06 2008
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CROSSREFS
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See A057137 for another version.
Cf. A033307.
A053064. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 10 2008]
Sequence in context: A014824 A060555 A138957 this_sequence A057932 A132943 A159901
Adjacent sequences: A007905 A007906 A007907 this_sequence A007909 A007910 A007911
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KEYWORD
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nonn,base
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AUTHOR
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R. Muller
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