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A007908 Concatenation of the numbers from 1 to n. 98
1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, 12345678910111213, 1234567891011121314, 123456789101112131415, 12345678910111213141516, 1234567891011121314151617 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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, May 29 2008, Paolo P. Lava, Jun 06 2008

Number of digits: A058183(n) = A055642(a(n)); sums of digits: A037123(n) = A007953(a(n)). - Reinhard Zumkeller, 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

Stephan finds no primes in the first 839 terms. I checked that there are no primes in the first 5000 terms. Heuristically there are infinitely many, about 0.5 log log n through the n-th term. - Charles R Greathouse IV, Sep 19 2012 [Expanded search to 20,000 without finding any primes. - Charles R Greathouse IV, Apr 17 2014]  [Independent search extended to 64,000 terms without finding any primes. - Dana Jacobsen, Apr 25 2014]

Early bird numbers for n > 1: a(2) = A116700(1) = 12; a(3) = A116700(52) = 123; a(4) = A116700(725) = 1234; a(5) = A116700(8074) = 12345; a(6) = A116700(85846) = 123456. - Reinhard Zumkeller, Dec 13 2012

For n < 10^4, a(n)/A000217(n) is an integer for n = 1, 2, and 5. The integers are 1, 4, and 823 (a prime), respectively. - Derek Orr, Sep 04 2014

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, A3.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..100

Y. Guo and M. Le, Smarandache concatenated power decimals and their irrationality, Smarandache Notions Journal, Vol. 9, No. 1-2. 1998, 100-102.

Clifford Pickover, Triangle of the Gods

F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.

R. W. Stephan, Factors and primes in two Smarandache sequences

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

FORMULA

a(n) = a(n-1)*10^floor[log10(10*n)]+n. - Paolo P. Lava, Feb 01 2008

a(n) = n+a(n-1)*10^A055642(n). - R. J. Mathar, 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 P. Lava, Jun 06 2008

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, 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 P. Lava, Jun 06 2008

MATHEMATICA

f[n_] := Block[{c = 0, k = 1}, While[k <= n, c = 10^Floor[1 + Log10[k]] c + k; k++]; c]; Array[f, 17] (* Robert G. Wilson v, Jun 24 2012 *)

Table[FromDigits[Flatten[IntegerDigits[Range[n]]]], {n, 20}] (* Alonso del Arte, Sep 19 2012 *)

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 P. Lava, Jun 06 2008

(PARI) a(n)=my(s=""); for(k=1, n, s=Str(s, k)); eval(s) \\ Charles R Greathouse IV, Sep 19 2012

(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 */

(Haskell)

a007908 = read . concatMap show . enumFromTo 1 :: Integer -> Integer

-- Reinhard Zumkeller, Dec 13 2012

CROSSREFS

See A057137 for another version.

Cf. A033307, A053064.

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, Aug 11 2010]

Sequence in context: A014824 A060555 A138957 * A057932 A132943 A187871

Adjacent sequences:  A007905 A007906 A007907 * A007909 A007910 A007911

KEYWORD

nonn,base,easy

AUTHOR

R. Muller

STATUS

approved

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Last modified October 24 23:54 EDT 2014. Contains 248516 sequences.