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A007908
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Concatenation of the numbers from 1 to n.
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89
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1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, 12345678910111213, 1234567891011121314, 123456789101112131415, 12345678910111213141516, 1234567891011121314151617
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also called the triangle of the gods (see Pickover link).
Sometimes called Smarandache consecutive numbers.
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
Number of digits: A058183(n)=A055642(a(n)); sums of digits: A037123(n)=A007953(a(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 10 2010]
Charles Nicol and John Selfridge ask if there are infinitely many primes in this sequence, see the Guy reference. [Charles R Greathouse IV, Dec 14 2011]
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REFERENCES
| 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
| 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, Consecutive Number Sequences
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FORMULA
| a(n)=a(n-1)*10^floor[log10(10*n)]+n - Paolo P. Lava (paoloplava(AT)gmail.com), 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
| 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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MATHEMATICA
| If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 10]]]; Table[AppendTo[n, IntegerDigits[w, 10]]; n=Flatten[n]; FromDigits[n, 10], {w, STARTPOINT, ENDPOINT}] [From Dylan Hamilton (Phalarisbull(AT)gmail.com), Aug 11 2010]
Join[{a=1}, Table[d=Length[IntegerDigits[n]]; a=a*10^d+n, {n, 2, 30}]]
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PROG
| (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
(MAGMA) [Seqint(Reverse(&cat[Reverse(Intseq(k)): k in [1..n]])): n in [1..17]]; // Bruno Berselli, May 27 2011
(Maxima) a[1]:1$ a[n]:=a[n-1]*10^floor(log(10*n)/log(10))+n$ makelist(a[n], n, 1, 17); [Bruno Berselli, May 27 2011]
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CROSSREFS
| See A057137 for another version.
Cf. A033307.
A053064. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 10 2008]
Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: this sequence, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447 [From Dylan Hamilton (Phalarisbull(AT)gmail.com), Aug 11 2010]
Sequence in context: A014824 A060555 A138957 * A057932 A132943 A187871
Adjacent sequences: A007905 A007906 A007907 * A007909 A007910 A007911
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KEYWORD
| nonn,base
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AUTHOR
| R. Muller
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