

A007908


Concatenation of the numbers from 1 to n.


117



1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, 12345678910111213, 1234567891011121314, 123456789101112131415, 12345678910111213141516, 1234567891011121314151617
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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 nth 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]
A note on heuristics: I wrote a quick program to count primes in sequences which are like A007908 but start at k instead of 1. I ran this for k = 1 to 100 and counted the primes up to 1000 (1000 possibilities for k = 1, 999 for k = 2, etc. up to 901 for k = 100). I then compared this to the expected count which is 0 if the number N is divisible by 2, 3, or 5 and 15/(4 log N) otherwise. (If N < 43 I counted the number as 1 instead.) k = 1 has 1.788 expected primes but only 0 actual (of course). k = 2 has 2.268 expected but 4 actual (see A262571, A089987). In total the expectation is 111.07 and the actual count is 110, well within the expected error of +/ 10.5.  Charles R Greathouse IV, Sep 28 2015
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 < 10^6, 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; Max Alekseyev, Sep 30 2015
In order to be a prime, a(n) must end in a digit 1, 3, 7 or 9, so only 4 among 10 consecutive values can be prime. (But a(64000) already has A058183(64000) > 300000 digits.) Also, a(64001) and a(64011) and more generally a(64001+10k) is divisible by 3 unless k == 2 (mod 3), but for k = 2, 5, 8, ... 23 these are divisible by small primes < 999. a(64261) is the first serious candidate in this subsequence.  M. F. Hasler, Sep 30 2015
There are no primes in the first 77000 terms.  Max Alekseyev, Oct 03 2015


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. 12. 1998, 100102.
Clifford Pickover, Triangle of the Gods
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., PhoenixChicago, 1993.
R. W. Stephan, Factors and primes in two Smarandache sequences
Eric Weisstein's World of Mathematics, Consecutive Number Sequences


FORMULA

a(n) = a(n1)*10^floor[log10(10*n)]+n.  Paolo P. Lava, Feb 01 2008
a(n) = n+a(n1)*10^A055642(n).  R. J. Mathar, May 31 2008
a(n) = prod(x,1,n,10^floor(1+log(10)^(1)*log(x))) * sum(y,1,n,prod(z,1,y,10^floor(log(10)^(1)*(log(10)+log(z))))^(1)*y).  Alexander R. Povolotsky and Paolo P. Lava, Jun 06 2008
a(n) = floor(C*10^(A058183(n))) with C the Champernowne constant, 0.123456789101112131415..., A033307.  José de Jesús Camacho Medina, Aug 19 2015


MAPLE

A055642 := proc(n) max(1, ilog10(n)+1) ; end: A007908 := proc(n) if n = 1 then 1; else A007908(n1)*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[n1]*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
(PARI) A007908(n, a=0)={for(d=1, #Str(n), my(t=10^d); for(k=t\10, min(t1, n), a=a*t+k)); a} \\ M. F. Hasler, Sep 30 2015


CROSSREFS

See A057137 for another version.
Cf. A033307, A053064, A000422 (left concatenations)
If we concatenate 1 through n but leave out k, we get sequences A262571 (leave out 1) through A262582 (leave out 12), etc., and again we can ask for the smallest prime in each sequence. See A262300 for a summary of these results. Primes seem to exist if we search far enough.  N. J. A. Sloane, Sep 29 2015
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]
Cf. also A089987.
Sequence in context: A014824 A060555 A138957 * A262582 A262581 A262580
Adjacent sequences: A007905 A007906 A007907 * A007909 A007910 A007911


KEYWORD

nonn,base,easy,changed


AUTHOR

R. Muller


STATUS

approved



