OFFSET
1,3
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
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.
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
EXAMPLE
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.
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 *)
PROG
(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
(Haskell)
a072762 n = foldl (\v d -> 2*v + d) 0 $ map a010051 [1..n]
-- Reinhard Zumkeller, Sep 17 2011
CROSSREFS
KEYWORD
nonn,nice,base
AUTHOR
Reinhard Zumkeller, Aug 08 2002
STATUS
approved