

A007908


Triangle of the gods: to get a(n), concatenate the decimal numbers 1,2,3,...,n.


189



1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, 12345678910111213, 1234567891011121314, 123456789101112131415, 12345678910111213141516, 1234567891011121314151617, 123456789101112131415161718
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OFFSET

1,2


COMMENTS

For the name "triangle of the gods" see Pickover link.  N. J. A. Sloane, Dec 15 2019
Lim_{n>infinity} a(n)/(Product_{i=1..n} 10^floor(1+(log(i)/log(10)))) yields the Champernowne constant (see A033307).  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 20000 without finding any primes.  Charles R Greathouse IV, Apr 17 2014] [Independent search extended to 64000 terms without finding any primes.  Dana Jacobsen, Apr 25 2014]
Elementary congruence arguments show that primes can occur only at indices congruent to 1, 7, 13, or 19 mod 30.  Roderick MacPhee, Oct 05 2015
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 10^5 terms.  Max Alekseyev, Oct 03 2015; Oct 11 2015
There are no primes in the first 200000 terms.  Serge Batalov, Oct 24 2015
There is a distributed project for continued search, using PRPNet/PFGW software; see the Mersenne Forum link below. Serge Batalov, Oct 18 2015
It appears that the Mersenne Forum search reached n = 344869 without finding a prime, and was then abandoned. It would be nice if someone could recover the final version of that link from the Wayback machine  the Great Smarandache PRPrime search, http://99.121.249.54:1200  so that we have a record of how far they searched.  N. J. A. Sloane, Apr 09 2018
The web page https://www.mersenneforum.org/showthread.php?t=20527&page=9 has a comment from Serge Balatov that seems to say that the search reached 10^6 without finding a prime. It would be nice to have this confirmed, and to get more details about how it was done.  N. J. A. Sloane, Dec 15 2019
The expected number of primes among the first million terms is about 0.6.  Ernst W. Mayer, Oct 09 2015
A few semiprimes exist among the early terms, but then become scarce: see A046461. For the base2 analog of this sequence (A047778), there is a 15decimal digit prime, but Hans Havermann has shown that the second prime would have more than 91000 digits.  N. J. A. Sloane, Oct 08 2015


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, Section A3, page 15, of 3rd edition, Springer, 2010.


LINKS

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..300 (First 100 terms from T. D. Noe)
Great Smarandache PRPrime search, Current status of search for a prime in this sequence [Broken link? It appears that this search reached n=344869 without finding a prime, and was then abandoned.  N. J. A. Sloane, Apr 09 2018]]
Y. Guo and M. Le, Smarandache concatenated power decimals and their irrationality, Smarandache Notions Journal, Vol. 9, No. 12. 1998, 100102.
Brady Haran and Neil Sloane, The Most Wanted Prime Number, Numberphile video (2021).
Ernst W. Mayer and others, Expected number of primes in OEIS A007908, Mersenne Forum, initial posting Oct 08 2015.
Mersenne Forum, Smarandache prime(s).
Clifford Pickover, Triangle of the Gods
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., PhoenixChicago, 1993.
R. W. Stephan, Factors and primes in two Smarandache sequences, viXra:1005.0104, 2011.
Bertrand Teguia Tabuguia, Explicit formulas for concatenations of arithmetic progressions, arXiv:2201.07127 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Smarandache Prime


FORMULA

a(n) = a(n1)*10^floor(log_10(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) = 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
# third Maple program:
a:= proc(n) a(n):= `if`(n=0, 0, parse(cat(a(n1), n))) end:
seq(a(n), n=1..22); # Alois P. Heinz, Jan 12 2021


MATHEMATICA

Table[FromDigits[Flatten[IntegerDigits[Range[n]]]], {n, 20}] (* Alonso del Arte, Sep 19 2012 *)
FoldList[#2 + #1 10^IntegerLength[#2] &, Range[20]] (* Eric W. Weisstein, Nov 06 2015 *)
FromDigits /@ Flatten /@ IntegerDigits /@ Flatten /@ Rest[FoldList[List, {}, Range[20]]] (* Eric W. Weisstein, Nov 04 2015 *)
FromDigits /@ Flatten /@ IntegerDigits /@ Rest[FoldList[Append, {}, Range[20]]] (* Eric W. Weisstein, Nov 04 2015 *)


PROG

(PARI) a(n)=my(s=""); for(k=1, n, s=Str(s, k)); eval(s) \\ Charles R Greathouse IV, Sep 19 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
(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
(Python)
def a(n): return int("".join(map(str, range(1, n+1))))
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Jan 12 2021


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.  Dylan Hamilton, Aug 11 2010
Entries that give the primes in sequences of this type: A089987, A262298, A262300, A262552, A262555.
For semiprimes see A046461.
See also A007376 (the almostnatural numbers), A071620 (primes in that sequence).
See also A033307 (the Champernowne constant) and A176942 (the Champernowne primes). A262043 is a variant of the present sequence.
A002782 is an amusing cousin of this sequence.
Least prime factor: A075019.
Sequence in context: A014824 A060555 A138957 * A262582 A262581 A262580
Adjacent sequences: A007905 A007906 A007907 * A007909 A007910 A007911


KEYWORD

nonn,base,easy


AUTHOR

R. Muller


EXTENSIONS

Name edited by N. J. A. Sloane, Dec 15 2019


STATUS

approved



