A139102 Numbers whose binary representation shows the distribution of prime numbers up to the n-th prime minus 1, using "0" for primes and "1" for nonprime numbers.

1, 2, 9, 37, 599, 2397, 38359, 153437, 2454999, 157119967, 628479869, 40222711647, 643563386359, 2574253545437, 41188056726999, 2636035630527967, 168706280353789919, 674825121415159677, 43188807770570219359, 691020924329123509751, 2764083697316494039005

Offset: 1

Author

Omar E. Pol, Apr 08 2008

Keywords

• nonn
• base

Comments

a(n) is the decimal representation of A139101(n) interpreted as binary number.

Examples

			a(4)=37 because 37 written in base 2 is 100101 and the string "100101" shows the distribution of prime numbers up to the 4th prime minus 1, using "0" for primes and "1" for nonprime numbers.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..468
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Crossrefs

Subset of A118255.

Cf. A000040, A018252, A139101, A139103, A139104, A139119, A139120, A139122.

Programs

Maple

A139101 := proc(n) option remember ; local a,p; if n = 1 then RETURN(1); else a := 10*A139101(n-1) ; for p from ithprime(n-1)+1 to ithprime(n)-1 do a := 10*a+1 ; od: fi ; RETURN(a) ; end: # R. J. Mathar, Apr 25 2008
bin2dec := proc(n) local nshft ; nshft := convert(n,base,10) ; add(op(i,nshft)*2^(i-1),i=1..nops(nshft) ) ; end: # R. J. Mathar, Apr 25 2008
A139102 := proc(n) bin2dec(A139101(n)) ; end: # R. J. Mathar, Apr 25 2008
seq(A139102(n),n=1..35) ; # R. J. Mathar, Apr 25 2008

Mathematica

Table[ sum = 0; For[i = 1, i <= Prime[n] - 1 , i++, sum = sum*2;
If[! PrimeQ[i], sum++]]; sum, {n, 1, 25}] (* Robert Price, Apr 03 2019 *)

PARI

a(n) = fromdigits(vector(prime(n)-1, k, !isprime(k)), 2); \\ Michel Marcus, Apr 04 2019

Python

from sympy import isprime, prime
def a(n):
    return int("".join(str(1-isprime(i)) for i in range(1, prime(n))), 2)
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Jan 10 2022

Python

# faster version for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    an = 0
    for k in count(1):
        an = 2 * an + int(not isprime(k))
        if isprime(k+1):
            yield an
print(list(islice(agen(), 21))) # Michael S. Branicky, Jan 10 2022

Formula

a(n) = A139104(n)/2.

Extensions

More terms from R. J. Mathar, Apr 25 2008

a(20)-a(21) from Robert Price, Apr 03 2019