A263856 -id:A263856 - cp's OEIS frontend

A264647 Smallest number m such that A263856(m) = n.

1, 2, 6, 4, 10, 8, 79, 199, 9, 108, 11, 29, 17, 15, 40, 80, 20, 59, 306, 22, 169, 38, 27, 82, 287, 41, 49, 209, 47, 135, 31, 36, 127, 112, 123, 162, 46, 89, 63, 54, 581, 43, 56, 770, 67, 48, 134, 52, 142, 69, 58, 101, 382, 466, 75, 64, 273, 95, 117, 126, 72

Offset: 1

Reinhard Zumkeller, Nov 19 2015

nonn

a(6949) > 22700000 if it exists. - Chai Wah Wu, Feb 01 2016

Reinhard Zumkeller and Chai Wah Wu, Table of n, a(n) for n = 1..6948 n = 1..391 from Reinhard Zumkeller

Cf. A263856.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a264647 = (+ 1) . fromJust . (`elemIndex` a263856_list)

f[p_] := f[p] = StringJoin @@ ToString /@ Reverse[IntegerDigits[p, 2]];
S[n_] := S[n] = SortBy[Prime[Range[n]], f];
A263856[n_] := A263856[n] = FirstPosition[S[n], Prime[n]][[1]];
a[n_] := a[n] = For[m = 1, True, m++, If[A263856[m] == n, Return[m]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 391}] (* Jean-François Alcover, Sep 22 2021 *)

A263856(a(n)) = n and A263856(m) != n for m < a(n).

A264596 Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 3, 7, 1, 6, 4, 10, 3, 10, 7, 15, 1, 10, 6, 16, 4, 15, 10, 22, 3, 16, 10, 24, 7, 22, 15, 31, 1, 18, 10, 28, 6, 25, 16, 36, 4, 25, 15, 37, 10, 33, 22, 46, 3, 28, 16, 42, 10, 37, 24, 52, 7, 36, 22, 52, 15, 46, 31, 63, 1, 34, 18, 52, 10, 45, 28

Offset: 0

N. J. A. Sloane, Nov 19 2015

			S_0 = [0], a(0) = 0;
S_1 = [0, 1], a(1) = 1;
S_2 = [0, 01, 1], a(2) = 1;
S_3 = [0, 01, 1, 11], a(3) = 3;
S_4 = [0, 001, 01, 1, 11], a(4) = 1;
S_5 = [0, 001, 01, 1, 101, 11], a(5) = 4;
S_6 = [0, 001, 01, 011, 1, 101, 11], a(6) = 3;
S_7 = [0, 001, 01, 011, 1, 101, 11, 111], a(7) = 7;
S_8 = [0, 0001, 001, 01, 011, 1, 101, 11, 111], a(8) = 1;
...

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Suggested by John Bodeen's A263856.

Cf. A188215.

a:= proc(n) option remember; `if`(n=0, 0,
      `if`(irem(n, 2, 'r')=0, a(r), a(r)+r+1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 19 2015

A264596[0]=0;A264596[n_]:=A264596[n]=A264596[Floor[n/2]]+Boole[OddQ[n]](Floor[n/2]+1);Array[A264596,100,0] (* Paolo Xausa, Nov 04 2023, after Alois P. Heinz *)

def A264596(n):
    return sorted(format(i,'b')[::-1] for i in range(n+1)).index(format(n,'b')[::-1]) # Chai Wah Wu, Nov 22 2015

a(2^n) = 1.
a(2^n-1) = 2^n-1.
a(2n) = a(n), a(2n+1) = a(n) + n+1, a(0) = 0. - Alois P. Heinz, Nov 19 2015
Conjecture: a(n) = n*(n + 3)/2 - A007814(A293290(n)) for n > 0. - Velin Yanev, Sep 12 2017

More terms from Alois P. Heinz, Nov 19 2015

A264662 Triangle read by rows: row n contains the first n primes in lexicographical order of their mirrored binary representation.

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 2, 5, 3, 7, 2, 5, 3, 11, 7, 2, 5, 13, 3, 11, 7, 2, 17, 5, 13, 3, 11, 7, 2, 17, 5, 13, 3, 19, 11, 7, 2, 17, 5, 13, 3, 19, 11, 7, 23, 2, 17, 5, 13, 29, 3, 19, 11, 7, 23, 2, 17, 5, 13, 29, 3, 19, 11, 7, 23, 31, 2, 17, 5, 37, 13, 29, 3, 19, 11

Offset: 1

Reinhard Zumkeller, Nov 20 2015

T(n,A263856(n)) = A000040(n): A263856(n) = index of prime(n) in n-th row.

			.   n |   T(n,k), k=1..n
. ----+-----------------------------------------------------------------
.   1 | 2                          01
.   2 | 2  3                       01 11
.   3 | 2  5  3                    01 101   11
.   4 | 2  5  3  7                 01 101   11   111
.   5 | 2  5  3 11  7              01 101   11   1101 111
.   6 | 2  5 13  3 11  7           01 101   1011 11   1101 111
.   7 | 2 17  5 13  3 11  7        01 10001 101  1011 11   1101  111
.   8 | 2 17  5 13  3 19 11  7     01 10001 101  1011 11   11001 1101 111
.   9 | 2 17  5 13  3 19 11  7 23
.  10 | 2 17  5 13 29  3 19 11  7 23
.  11 | 2 17  5 13 29  3 19 11  7 23 31
.  12 | 2 17  5 37 13 29  3 19 11  7 23 31
.  13 | 2 17 41  5 37 13 29  3 19 11  7 23 31
.  14 | 2 17 41  5 37 13 29  3 19 11 43 7  23 31
.  15 | 2 17 41  5 37 13 29  3 19 11 43  7 23 47 31
.  16 | 2 17 41  5 37 53 13 29  3 19 11 43  7 23 47 31
.  17 | 2 17 41  5 37 53 13 29  3 19 11 43 59  7 23 47 31
.  18 | 2 17 41  5 37 53 13 29 61  3 19 11 43 59  7 23 47 31
.  19 | 2 17 41  5 37 53 13 29 61  3 67 19 11 43 59  7 23 47 31
.  20 | 2 17 41  5 37 53 13 29 61  3 67 19 11 43 59  7 71 23 47 31

Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened

Cf. A263846, A000040, A007088, A007504 (row sums), A264666 (partial row products), A037126 (rows sorted naturally).

import Data.List (inits, sortBy); import Data.Function (on)
a264662 n k = a264662_tabl !! (n-1) !! (n-1)
a264662_row n = a264662_tabl !! (n-1)
a264662_tabl = map (sortBy (compare `on` (reverse . show . a007088))) $
                   tail $ inits a000040_list

row[n_] := SortBy[Prime[Range[n]], StringJoin[ToString /@ Reverse[IntegerDigits[#, 2]]]&];
Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Sep 25 2021 *)

cp's OEIS Frontend

A264647 Smallest number m such that A263856(m) = n.

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A264596 Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.

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A264662 Triangle read by rows: row n contains the first n primes in lexicographical order of their mirrored binary representation.

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Haskell

Mathematica