A007908 -id:A007908 - cp's OEIS frontend

A173426 a(n) is obtained by starting with 1, sequentially concatenating all decimal numbers up to n, and then, starting from n-1, sequentially concatenating all decimal numbers down to 1.

1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 12345678910987654321, 123456789101110987654321, 1234567891011121110987654321, 12345678910111213121110987654321, 123456789101112131413121110987654321

Offset: 1

Umut Uludag, Feb 18 2010

The first prime in this sequence is the 20-digit number a(10) = 12345678910987654321. On Jul 20 2015, Shyam Sunder Gupta reported on the Number Theory Mailing List that he has found what is probably the second prime in the sequence. This is the 2446th term, namely the 17350-digit probable prime 1234567..244524462445..7654321. See A359148. - N. J. A. Sloane, Jul 29 2015 - Aug 03 2015
There are no other (PR)prime members in this sequence for n<60000. - Serge Batalov, Jul 29 2015
David Broadhurst gives heuristic arguments which suggest that this sequence contains infinitely many primes.
See A075023 and A075024 for the smallest and largest prime factor of the terms. - M. F. Hasler, Jul 29 2015
Using summation in decimal length clades, one can obtain analytical expressions for the sequence:
a(n) = A002275(n)^2, for 1 <= n < 10;
a(n) = (120999998998*10^(4*n-28) - 2*10^(2*n-9) + 8790000000121)/99^2, for 10 <= n < 10^2;
a(n) = (120999998998*10^(6*n-227) - (1099022*10^(6*n-406) + 242*10^(3*n-108) - 1087789*10^191)/111^2 + 8790000000121)/99^2, for 10^2 <= n < 10^3; etc. - Serge Batalov, Jul 29 2015
Curiously, 1234567891010987654321 is also a prime (see A259937). - N. J. A. Sloane, Nov 30 2021

D. Broadhurst, Primes from concatenation: results and heuristics, Number Theory List, Aug 01 2015 and later postings.

G. C. Greubel, Table of n, a(n) for n = 1..150
FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.
Shyam Sunder Gupta, Puzzle 794. Prime Generalized Palindromes, The Prime Puzzles and Problems Connection.
S. S. Gupta, A new 17350 digit Symmetric Prime, NmbrThry List, July 20, 2015.
Brady Haran and N. J. A. Sloane, The Most Wanted Prime Number, Numberphile series on YouTube, Dec 15 2021.
Bertrand Teguia Tabuguia, Explicit formulas for concatenations of arithmetic progressions, arXiv:2201.07127 [math.CO], 2022.
Index entries for sequences related to Most Wanted Primes video

This sequence and A002477 (Wonderful Demlo numbers) agree up to the 9th term.

Cf. A002275, A007908, A075023, A075024, A259937, A359148.

a:= n-> parse(cat($1..n, n-i$i=1..n-1)):
seq(a(n), n=1..14);  # Alois P. Heinz, Dec 01 2021

Table[FromDigits[Flatten[IntegerDigits/@Join[Range[n],Reverse[Range[ n-1]]]]],{n,15}] (* Harvey P. Dale, Sep 02 2015 *)

A173426(n)=eval(concat(vector(n*2-1,k,if(kM. F. Hasler, Jul 29 2015

def A173426(n): return int(''.join(str(d) for d in range(1,n+1))+''.join(str(d) for d in range(n-1,0,-1))) # Chai Wah Wu, Dec 01 2021

a(n) = concatenate(1,2,3,...,n-2,n-1,n,n-1,n-2,...,3,2,1).

More terms from and minor edits by M. F. Hasler, Jul 29 2015

A037123 a(n) = a(n-1) + sum of digits of n.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 100, 102, 105, 109, 114, 120, 127, 135, 144, 154, 165, 168, 172, 177, 183, 190, 198, 207, 217, 228, 240, 244, 249, 255, 262, 270, 279, 289, 300, 312, 325, 330, 336, 343, 351, 360, 370, 381

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

Sum of digits of A007908(n). - Franz Vrabec, Oct 22 2007
Also digital sum of A138793(n) for n > 0. - Bruno Berselli, May 27 2011
Sum of the digital sum of i for i from 0 to n. - N. J. A. Sloane, Nov 13 2013

N. Agronomof, Sobre una función numérica, Revista Mat. Hispano-Americana 1 (1926), 267-269.
Maurice d'Ocagne, Sur certaines sommations arithmétiques, J. Sciencias Mathematicas e Astronomicas 7 (1886), 117-128.

David A Corneth, Table of n, a(n) for n = 0..10008
P.-H. Cheo and S.-C. Yien, A problem on the k-adic representation of positive integers, Acta Math. Sinica 5, 433-438 (1955).
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
H. Delange, Sur la fonction sommatoire de la fonction "somme des chiffres", Enseignement Math. (2) 21 (1975), 31-47.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 263-271.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
H. Riede, Asymptotic estimation of a sum of digits, Fibonacci Q. 36, No. 1, 72-75 (1998).
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A004207, A016052, A131383, A131384, A131451.
Cf. also A074784, A231688, A231689.
Partial sums of A007953.

[ n eq 0 select 0 else &+[&+Intseq(k): k in [0..n]]: n in [0..56] ];  // Bruno Berselli, May 27 2011

# From N. J. A. Sloane, Nov 13 2013:
digsum:=proc(n,B) local a; a := convert(n, base, B):
add(a[i], i=1..nops(a)): end;
f:=proc(n,k,B) global digsum; local i;
add( digsum(i,B)^k,i=0..n); end;
lprint([seq(digsum(n,10),n=0..100)]); # A007953
lprint([seq(f(n,1,10),n=0..100)]); #A037123
lprint([seq(f(n,2,10),n=0..100)]); #A074784
lprint([seq(f(n,3,10),n=0..100)]); #A231688
lprint([seq(f(n,4,10),n=0..100)]); #A231689

Table[Plus@@Flatten[IntegerDigits[Range[n]]], {n, 0, 200}] (* Enrique Pérez Herrero, Oct 12 2015 *)
a[0] = 0; a[n_] := a[n - 1] + Plus @@ IntegerDigits@ n; Array[a, 70, 0] (* Robert G. Wilson v, Jul 06 2018 *)

a(n)=n*(n+1)/2-9*sum(k=1,n,sum(i=1,ceil(log(k)/log(10)),floor(k/10^i)))

a(n)={n++;my(t,i,s);c=n;while(c!=0,i++;c\=10);for(j=1,i,d=(n\10^(i-j))%10;t+=(10^(i-j)*(s*d+binomial(d,2)+d*9*(i-j)/2));s+=d);t} \\ David A. Corneth, Aug 16 2013

for $i (0..100){ @j = split "", $i; for (@j){ $sum += $; } print "$sum,"; } __END_ # gamo(AT)telecable.es

a(n) = Sum_{k=0..n} s(k) = Sum_{k=0..n} A007953(k), where s(k) denote the sum of the digits of k in decimal representation. Asymptotic expression: a(n-1) = Sum_{k=0..n-1} s(k) = 4.5*n*log_10(n) + O(n). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
a(n) = n*(n+1)/2 - 9*Sum_{k=1..n} Sum_{i=1..ceiling(log_10(k))} floor(k/10^i). - Benoit Cloitre, Aug 28 2003
From Hieronymus Fischer, Jul 11 2007: (Start)
G.f.: Sum_{k>=1} ((x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k)))/(1-x)^2.
a(n) = (1/2)*((n+1)*(n - 18*Sum_{k>=1} floor(n/10^k)) + 9*Sum_{k>=1} (1 + floor(n/10^k))*floor(n/10^k)*10^k).
a(n) = (1/2)*((n+1)*(2*A007953(n)-n) + 9*Sum_{k>=1} (1+floor(n/10^k))*floor(n/10^k)*10^k). (End)
a(n) = A007953(A053064(n)). - Reinhard Zumkeller, Oct 10 2008
From Wojciech Raszka, Jun 14 2019: (Start)
a(10^k - 1) = 10*a(10^(k - 1) - 1) + 45*10^(k - 1) for k > 0.
a(n) = a(n mod m) + MSD*a(m - 1) + (MSD*(MSD - 1)/2)*m + MSD*((n mod m) + 1), where m = 10^(A055642(n) - 1), MSD = A000030(n). (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

A047778 Concatenation of the first n numbers in binary (converted to base 10).

Original entry on oeis.org

1, 6, 27, 220, 1765, 14126, 113015, 1808248, 28931977, 462911642, 7406586283, 118505380540, 1896086088653, 30337377418462, 485398038695407, 15532737238253040, 497047591624097297, 15905522931971113522, 508976733823075632723, 16287255482338420247156

Offset: 1

Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)

The smallest prime in this sequence is 485398038695407. What is the full subsequence of primes? - N. J. A. Sloane, Oct 03 2015

There is only the one prime in the first 22400 terms, making a second prime > 10^91000. - Hans Havermann, Oct 07 2015

			a(4) = 1 10 11 100 [base 2] = 220 [base 10].

Joe B. Stephen, Table of n, a(n) for n = 1..400 (terms 1..250 from Reinhard Zumkeller)

Cf. A001855 (bit counts, offset by 1), A061168, A066716.

Concatenation of first n numbers in other bases: 2: this sequence, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

a047778 = (foldl (\v d -> 2*v + d) 0) . concatMap (reverse . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)) .
   enumFromTo 1
-- Reinhard Zumkeller, Feb 19 2012

conc:= (x,y) -> x*2^(1+ilog2(y))+y:
a[1]:= 1:
for n from 2 to 30 do a[n]:= conc(a[n-1],n) od:
seq(a[n],n=1..30); # Robert Israel, Oct 07 2015

If[STARTPOINT==1,n={},n=Flatten[IntegerDigits[Range[STARTPOINT-1],2]]]; Table[AppendTo[n,IntegerDigits[w,2]];n=Flatten[n];FromDigits[n,2],{w,STARTPOINT,ENDPOINT}] (* Dylan Hamilton, Aug 04 2010 *)
f[n_] := FromDigits[ Flatten@ IntegerDigits[ Range@n, 2], 2]; Array[f, 18] (* Robert G. Wilson v, Nov 07 2010 *)
Module[{n = 1}, NestList[#*2^BitLength[++n] + n &, 1, 25]] (* Paolo Xausa, Sep 30 2024 *)

cb(a,b)=a<<#binary(b) + b
a(n)=fold(cb, [1..n]) \\ Charles R Greathouse IV, Jun 21 2017

A047778_vec(N=20,s)=vector(N,k,s=s<M. F. Hasler, Oct 25 2019

def a(n): return int("".join([(bin(i))[2:] for i in range(1, n+1)]), 2)
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Jan 06 2021

from functools import reduce
def A047778(n): return reduce(lambda i,j:(i<Chai Wah Wu, Feb 26 2023

a(n) = a(n-1)*2^(1+floor(log_2(n))) + n. - Henry Bottomley, Jan 12 2001

a(n) = 4C / 2^frac(log_2(n)) * n^{n+1} / r(frac(log_2(n)))^n + O(1), where r(x) = 2^{x - 1 + 2^{1-x}}; frac is the fractional part function frac(x) = x - floor(x); and C is the binary Champernowne constant (A066716). (In fact, a(n) is the floor of this expression; the error term is between 1/2 and 1.) r(x) takes on values between e*log(2) and 2 for x in the range 0 to 1. It follows using Stirling's approximation that the radius of convergence for the e.g.f. is log 2. - Franklin T. Adams-Watters, Sep 07 2006

More terms from Patrick De Geest, May 15 1999

Name edited by Joe B. Stephen, Jul 22 2023

A058183 Number of digits in concatenation of first n positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 1

Henry Bottomley, Nov 17 2000

Or, total number of digits in numbers from 1 through n.

			a(12) = 15 since 123456789101112 has 15 digits.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
GeeksforGeeks, Count total number of digits from 1 to n
Eric Weisstein's World of Mathematics, Smarandache Number

Cf. A000422, A007908, A053064, A055642.

a:= proc(n) a(n):= `if`(n=0, 0, a(n-1) +length(n)) end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2013
a := proc(n) local d; d:=floor(log10(n))+1; (n+1)*d - (10^d-1)/9; end; # N. J. A. Sloane, Feb 20 2020

Length/@ Flatten/@ IntegerDigits/@ Flatten/@ Rest[FoldList[List, {}, Range[70]]] (* Eric W. Weisstein, Nov 04 2015 *)
Table[With[{d = IntegerLength[n]}, (n+1) d - (10^d -1)/9], {n, 70}] (* Eric W. Weisstein, Nov 06 2015 *)
IntegerLength/@ FoldList[#2 + #1 10^IntegerLength[#2] &, Range[70]] (* Eric W. Weisstein, Nov 06 2015 *)
Accumulate[ IntegerLength@ # & /@ Range @ 70] (* Robert G. Wilson v, Jul 31 2018 *)

a(n)=my(t=log(10*n+.5)\log(10));n*t+t-10^t\9 \\ Charles R Greathouse IV, Sep 19 2012

a(n) = sum(k=1, n, #digits(k)); \\ Michel Marcus, Jan 01 2017

def A058183(n): return (n+1)*(s:=len(str(n))) - (10**s-1)//9 # Chai Wah Wu, May 02 2023

a(n) = (n+1)*floor(log_10(10*n)) - (10^floor(log_10(10*n))-1)/(10-1).
a(n) = a(n-1) + floor(log_10(10*n)).
a(n) = A055642(A007908(n)).
a(n) = A055642(A053064(n)). - Reinhard Zumkeller, Oct 10 2008
a(n) ~ n log_10 n + O(n). In particular lim inf (n log_10 n - a(n))/n = (1+log(10/9)+log(log(10)))/log(10) and the corresponding lim sup is 10/9. - Charles R Greathouse IV, Sep 19 2012
G.f.: (1-x)^(-2)*Sum_{j>=0} x^(10^j). - Robert Israel, Nov 04 2015
a(n) = b(n)*(n + 1) - (10^b(n) - 19)/9 - 2, where b(n) = A055642(n). - Lorenzo Sauras Altuzarra, May 09 2020
a(n) = A055642(A000422(n)). - Michel Marcus, Sep 11 2021

A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213.... Sequence gives numbers that occur in the string ahead of their natural place, sorted into increasing order (cf. A117804).

Original entry on oeis.org

12, 21, 23, 31, 32, 34, 41, 42, 43, 45, 51, 52, 53, 54, 56, 61, 62, 63, 64, 65, 67, 71, 72, 73, 74, 75, 76, 78, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 110, 111, 112, 121, 122, 123, 131, 132, 141, 142, 151, 152, 161, 162, 171

Offset: 1

Bernardo Recamán, Jul 22 2007

Based on an idea by Argentinian puzzle creator Jaime Poniachik, these numbers were introduced by Martin Gardner in 2005 in the magazine Math. Horizons, published by the MAA.
A048992 is a similar sequence, but is different because it does not contain 21, etc. - see comments in A048992.
A220376(n) = position of a(n) in 1234567891011121314151617181... . - Reinhard Zumkeller, Dec 13 2012

			"12" appears at the start of the string, ahead of its position after "11", so is a member.
So are 123, 23, 1234, 234, 34, ... and sorting these into increasing order we get 12, 21, 23, 31, ... - _N. J. A. Sloane_, Aug 28 2019

Martin Gardner, Transcendentals and early birds, Math. Horizons, XIII(2) (2005), pp. 5, 34 (published by Math. Assoc. America).

Joshua Zucker and R. Zumkeller, Table of n, a(n) for n = 1..10000 (first 675 terms from Joshua Zucker)
S. W. Golomb, Early Bird Numbers, Puzzle Column in IEEE Inform. Soc. Newsletter, 52(4) (2002), p. 10.
S. W. Golomb, Early Bird Numbers: Solutions, IEEE Inform. Soc. Newsletter, 53(1) (2003), p. 30.

Cf. A117804. A131881 gives complement.
Cf. A048991 and A048992 (Rollman numbers).
Cf. A007908 (subsequence, apart from initial 1).

s:= ""; for n:=1 to 200 do sn:=itoa(n);
if substr_index(s, sn) >= 0 then write(n, ","); end;
s:=concat(s, sn); end; (* Klaus Brockhaus, Jul 23 2007 *)

import Data.List (isPrefixOf, find)
import Data.Maybe (fromJust)
a116700 n = a116700_list !! (n-1)
a116700_list = filter early [1 ..] where
   early z = not (reverse (show (z - 1)) `isPrefixOf` fst bird) where
      bird = fromJust $ find ((show z `isPrefixOf`) . snd) xys
   xys = iterate (\(us, v : vs) -> (v : us, vs))
                 ([], concatMap show [0 ..])
-- Reinhard Zumkeller, Dec 13 2012

s = ""; Reap[For[n=1, n <= 200, n++, sn = ToString[n]; If[StringPosition[s, sn, 1] =!= {}, Sow[n]]; s = s <> sn]][[2, 1]] (* Jean-François Alcover, Nov 04 2016, after Klaus Brockhaus *)

def aupto(limit):
    s, alst = "", []
    for k in range(1, limit+1):
        sk = str(k)
        if sk in s: alst.append(k)
        s += sk
    return alst
print(aupto(171)) # Michael S. Branicky, Dec 21 2021

10 X=""
20 for N=1 to 396
30 A=cutspc(str(N))
40 if instr(X,A)>0 then print N;
50 X+=A
60 next N
70 'Warut Roonguthai, Jul 23 2007

Asymptotically, the early bird numbers have density 1 [Golomb].

More terms from Warut Roonguthai and Klaus Brockhaus, Jul 23 2007

Golomb links from Jeremy Gardiner, Jul 23 2007

A048435 Take the first n numbers written in base 3, concatenate them, then convert from base 3 to base 10.

Original entry on oeis.org

1, 5, 48, 436, 3929, 35367, 318310, 2864798, 77349555, 2088437995, 56387825876, 1522471298664, 41106725063941, 1109881576726421, 29966802571613382, 809103669433561330, 21845799074706155927, 589836575017066210047, 15925587525460787671288, 429990863187441267124796

Offset: 1

Patrick De Geest, May 15 1999

The first three primes in this sequence occur for n = 2 (a(2) = 5), n = 5 (a(5) = 3929), and n = 82 (a(82) = 1.1247...*10^140). - Kurt Foster, Oct 24 2015 [Comment added by N. J. A. Sloane, Oct 25 2015]

According to a comment made by Jeff Peltier following the "Most Wanted Prime" video, n = 2546 also gives a prime. See A360503. - N. J. A. Sloane, Feb 17 2023

			a(6): (1)(2)(10)(11)(12)(20) = 1210111220_3 = 35367.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Brady Haran and N. J. A. Sloane, Most Wanted Prime, Numberphile video, December 2021.
Index entries for sequences related to Most Wanted Primes video

Primes: A360503.

Concatenation of first n numbers in other bases: 2: A047778, 3: this sequence, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*3^(1+Ilog(3, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1,n={},n=Flatten[IntegerDigits[Range[STARTPOINT-1],3]]]; Table[AppendTo[n,IntegerDigits[w,3]];n=Flatten[n];FromDigits[n,3],{w,STARTPOINT,ENDPOINT}] (* Dylan Hamilton, Aug 09-04 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 3], 3]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A048436 Take the first n numbers written in base 4, concatenate them, then convert from base 4 to base 10.

Original entry on oeis.org

1, 6, 27, 436, 6981, 111702, 1787239, 28595832, 457533321, 7320533146, 117128530347, 1874056485564, 29984903769037, 479758460304606, 7676135364873711, 491272663351917520, 31441450454522721297, 2012252829089454163026, 128784181061725066433683

Offset: 1

Patrick De Geest, May 15 1999

There is no prime among the first 5000 terms (emails from Kurt Foster, Oct 21 2015 and Oct 24 2015). When is the first prime? - N. J. A. Sloane, Oct 25 2015

There is no prime among the first 45000 terms. - Giovanni Resta, Jun 07 2018

			a(7): (1)(2)(3)(10)(11)(12)(13) = 12310111213_4 = 1787239.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014825.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: this sequence, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447. - Dylan Hamilton, Aug 11 2010

[n eq 1 select 1 else Self(n-1) * 4^(1+Ilog(4,n)) + n: n in [1..20]]; // Jason Kimberley, Nov 27 2012

a[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 4], 4]; Array[a, 20] (* Vincenzo Librandi, Dec 30 2012 *)

from functools import reduce
def A048436(n): return reduce(lambda i,j:(i<<(bool((m:=j.bit_length())&1)<<1)+(m&-2))+j,range(n+1)) # Chai Wah Wu, Feb 26 2023

a(n) = a(n-1) * 4^(1 + floor(log4(n))) + n. [Moved from A117640 by Jason Kimberley, Nov 27 2012]

A262582 Concatenation of the numbers from 1 to n but omitting 12.

Original entry on oeis.org

1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101113, 12345678910111314, 1234567891011131415, 123456789101113141516, 12345678910111314151617, 1234567891011131415161718, 123456789101113141516171819, 12345678910111314151617181920

Offset: 1

N. J. A. Sloane, Sep 25 2015

The first (probable) prime in this sequence is a(960) = 1234567891011131415...961 (ending in 961), found by David Broadhurst, Sep 28 2015.

No other primes through a(10000). - Robert Price, Nov 04 2018

Cf. A007908, A262299, A262300, A262571-A262582.

See A262300 for more about this problem.

Module[{nn=30, c}, c=Drop[Range[nn], {12}]; Table[FromDigits[Flatten[IntegerDigits/@Take[c, n]]], {n, nn - 1}]] (* Vincenzo Librandi, Nov 05 2018 *)

A048437 Take the first n numbers written in base 5, concatenate them, then convert from base 5 to base 10.

Original entry on oeis.org

1, 7, 38, 194, 4855, 121381, 3034532, 75863308, 1896582709, 47414567735, 1185364193386, 29634104834662, 740852620866563, 18521315521664089, 463032888041602240, 11575822201040056016, 289395555026001400417, 7234888875650035010443, 180872221891250875261094

Offset: 1

Patrick De Geest, May 15 1999

The first three primes in this sequence occur for n = 2 (a(2) = 7), n = 113 (a(113) = 7.4484...*10^216), n = 162 (a(162) = 1.5188...*10^346). - Kurt Foster, Oct 24 2015 [Comment added by N. J. A. Sloane, Oct 25 2015]

			a(7) = 1 2 3 4 10 11 12 = 3034532_10.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014827.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: this sequence, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*5^(1+Ilog(5, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 5]]]; Table[AppendTo[n, IntegerDigits[w, 5]]; n=Flatten[n]; FromDigits[n, 5], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 5], 5]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A048447 Take the first n numbers written in base 16, concatenate them, then convert from base 16 to base 10.

Original entry on oeis.org

1, 18, 291, 4660, 74565, 1193046, 19088743, 305419896, 4886718345, 78187493530, 1250999896491, 20015998343868, 320255973501901, 5124095576030430, 81985529216486895, 20988295479420645136, 5373003642731685154833, 1375488932539311399637266, 352125166730063718307140115

Offset: 1

Patrick De Geest, May 15 1999

			a(16) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(C)(D)(E)(F)(10) = 123456789ABCDEF10_16 = 20988295479420645136.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014899.

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: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: this sequence.

[n eq 1 select 1 else Self(n-1)*16^(1+Ilog(16, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 16]]]; Table[AppendTo[n, IntegerDigits[w, 16]]; n=Flatten[n]; FromDigits[n, 16], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 16], 16]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

from functools import reduce
def A048447(n): return reduce(lambda i,j:(i<<(bool((m:=j.bit_length())&3)<<2)+(m&-4))+j,range(n+1)) # Chai Wah Wu, Feb 26 2023

cp's OEIS Frontend

A173426 a(n) is obtained by starting with 1, sequentially concatenating all decimal numbers up to n, and then, starting from n-1, sequentially concatenating all decimal numbers down to 1.

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A037123 a(n) = a(n-1) + sum of digits of n.

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A047778 Concatenation of the first n numbers in binary (converted to base 10).

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A058183 Number of digits in concatenation of first n positive integers.

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A116700 "Early bird" numbers: write the natural numbers in a string 12345678910111213.... Sequence gives numbers that occur in the string ahead of their natural place, sorted into increasing order (cf. A117804).

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