A072762 n coded as binary word of length=n with k-th bit set iff k is prime (1<=k<=n), decimal value.

0, 1, 3, 6, 13, 26, 53, 106, 212, 424, 849, 1698, 3397, 6794, 13588, 27176, 54353, 108706, 217413, 434826, 869652, 1739304, 3478609, 6957218, 13914436, 27828872, 55657744, 111315488, 222630977, 445261954, 890523909, 1781047818, 3562095636, 7124191272

Offset: 1

Author

Reinhard Zumkeller, Aug 08 2002

Keywords

• nonn
• nice
• base

Comments

a(n) is odd iff n is prime.

a(p) where p is prime is the numerator of Sum_{q <= p} 1/2^q where the sum is over primes up to p. - Alexander Adamchuk, Aug 22 2006

The n-th approximation to the Prime Constant is given by a(n)/2^n. - Anton Vrba (antonvrba(AT)yahoo.com), Nov 24 2006

Examples

			a(6) = '011010' = (((0*2+1)*2+1)*2*2+1)*2 = 26.
a(7) = '0110101' = (((0*2+1)*2+1)*2*2+1)*2*2+1 = 53.
a(8) = '01101010' = ((((0*2+1)*2+1)*2*2+1)*2*2+1)*2 = 106.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3323 (first 300 terms from T. D. Noe)
• Eric Weisstein's World of Mathematics, Prime Constant.

Crossrefs

Cf. A051006, A100634, A010051, A000720, A118255.

Programs

Haskell

a072762 n = foldl (\v d -> 2*v + d) 0 $ map a010051 [1..n]
-- Reinhard Zumkeller, Sep 17 2011

Maple

a:= proc(n) option remember;
      `if`(n<2, 0, 2 * a(n-1) + `if`(isprime(n), 1, 0))
    end:
seq(a(n), n=1..40);  #  Alois P. Heinz, Jan 18 2011

Mathematica

a[1] = 0; a[n_] := a[n] = 2*a[n-1] + Boole[PrimeQ[n]]; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Jun 14 2013 *)
nxt[{n_,a_}]:={n+1,Boole[PrimeQ[n+1]]+2a}; Transpose[NestList[nxt,{1,0},30]][[2]] (* Harvey P. Dale, Jan 07 2015 *)

PARI

an=0; print1(an,", "); for(n=2,31, an=2*an+isprime(n); print1(an,", ")) \\ Washington Bomfim, Jan 18 2011

PARI

a(n)=my(s=1,p=2);forprime(q=3,n,s=s<<(q-p)+1;p=q);s<<(n-p) \\ Charles R Greathouse IV, Jun 03 2013

Formula

a(1) = 0 and a(n) = a(n-1)*2 + A010051(n) for n>1.

a(n) = (1/2)*(pi(n) + Sum_{i=1..n} 2^(n-i)*pi(i)), where pi = A000720. - Ridouane Oudra, Aug 26 2019

a(n) = floor(c*2^n), where c = A051006 is the prime constant. - Lorenzo Sauras Altuzarra, Jan 03 2023