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A047778 Concatenation of first n numbers in binary. 28
1, 6, 27, 220, 1765, 14126, 113015, 1808248, 28931977, 462911642, 7406586283, 118505380540, 1896086088653, 30337377418462, 485398038695407, 15532737238253040, 497047591624097297, 15905522931971113522 (list; graph; refs; listen; history; text; internal format)
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

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

FORMULA

a(n) = a(n-1)*2^(1+floor[log2(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 = 220

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

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. - Dylan Hamilton, Aug 11 2010

Sequence in context: A223557 A060977 A267630 * A164985 A048436 A276412

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

STATUS

approved

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Last modified May 22 15:37 EDT 2017. Contains 286877 sequences.