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

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

Offset

1,2

Comments

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

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(Haskell)
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
(PARI) cb(a, b)=a<<#binary(b) + b
a(n)=fold(cb, [1..n]) \\ Charles R Greathouse IV, Jun 21 2017
(PARI) A047778_vec(N=20, s)=vector(N, k, s=s<<logint(k*2, 2)+k) \\ M. F. Hasler, Oct 25 2019
(Python)
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
(Python)
from functools import reduce
def A047778(n): return reduce(lambda i, j:(i<<j.bit_length())+j, range(n+1)) # Chai Wah Wu, Feb 26 2023

Crossrefs

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.

Sequence in context: A060977 A267630 A351737 * A290802 A360754 A367886

Adjacent sequences: A047775 A047776 A047777 * A047779 A047780 A047781

Keyword

easy,nonn,base,nice

Author

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

Extensions

More terms from Patrick De Geest, May 15 1999

Name edited by Joe B. Stephen, Jul 22 2023

Status

approved