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

Author

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

Keywords

• easy
• nonn
• base
• nice

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

Examples

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

Links

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

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.

Programs

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

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 *)
Module[{n = 1}, NestList[#*2^BitLength[++n] + n &, 1, 25]] (* Paolo Xausa, Sep 30 2024 *)

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<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<Chai Wah Wu, Feb 26 2023

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

Extensions

More terms from Patrick De Geest, May 15 1999

Name edited by Joe B. Stephen, Jul 22 2023