A263856 Let S_n be the list of the first n primes written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n-th prime in S_n.

1, 2, 2, 4, 4, 3, 2, 6, 9, 5, 11, 4, 3, 11, 14, 6, 13, 9, 11, 17, 3, 20, 14, 5, 2, 9, 23, 20, 12, 4, 31, 17, 5, 23, 12, 32, 17, 22, 32, 15, 26, 14, 42, 2, 11, 37, 29, 46, 27, 14, 9, 48, 6, 40, 2, 43, 22, 51, 18, 12, 43, 17, 39, 56, 14, 32, 45, 6, 50

Offset: 1

Author

John Bodeen, Oct 28 2015

Keywords

• nonn
• base

Comments

a(A264647(n)) = n and a(m) != n for m < A264647(n). - Reinhard Zumkeller, Nov 19 2015

A264662(n,a(n)) = A000040(n): a(n) = index of prime(n) in n-th row of triangle A264662. - Reinhard Zumkeller, Nov 20 2015

Examples

			S_1 = [01], a(1) = 1;
S_2 = [01, 11], a(2) = 2;
S_3 = [01, 101, 11], a(3) = 2;
S_4 = [01, 101, 11, 111], a(4) = 4;
S_5 = [01, 101, 11, 1101, 111], a(5) = 4;
S_5 = [01, 101, 1011, 11, 1101, 111], a(6) = 3;
...
		

Links

• John Bodeen, Table of n, a(n) for n = 1..24771
• John Bodeen, OCaml program to generate the sequence

Crossrefs

A004676 is the sequence upon which the lexicographic ordering is based.

Cf. A264596.

Cf. A264647, A264662.

Programs

Haskell

import Data.List (insertBy); import Data.Function (on)
import Data.List (elemIndex); import Data.Maybe (fromJust)
a263856 n = a263856_list !! (n-1)
a263856_list = f [] a004676_list where
   f bps (x:xs) = y : f bps' xs where
     y = fromJust (elemIndex x bps') + 1
     bps' = insertBy (compare `on` (reverse . show)) x bps
-- Reinhard Zumkeller, Nov 19 2015

Maple

s:= proc(n) s(n):= cat("", convert(ithprime(n), base, 2)[]) end:
a:= n-> ListTools[BinarySearch](sort([seq(s(i), i=1..n)]), s(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 19 2015

Mathematica

S[n_] := S[n] = SortBy[Prime[Range[n]], StringJoin @@ ToString /@ Reverse[IntegerDigits[#, 2]]&];
a[n_] := FirstPosition[S[n], Prime[n]][[1]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 22 2021 *)

Python

from sympy import prime
def A263856(n):
    return 1+sorted(format(prime(i),'b')[::-1] for i in range(1,n+1)).index(format(prime(n),'b')[::-1]) # Chai Wah Wu, Nov 22 2015