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A048672 Binary encoding of squarefree numbers (A005117): A048640(n)/2. 15
0, 1, 2, 4, 3, 8, 5, 16, 32, 9, 6, 64, 128, 10, 17, 256, 33, 512, 7, 1024, 18, 65, 12, 2048, 129, 34, 4096, 11, 8192, 257, 16384, 66, 32768, 20, 130, 513, 65536, 131072, 1025, 36, 19, 262144, 258, 13, 524288, 1048576, 2049, 24, 35, 2097152, 4097, 4194304, 68 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Permutation of nonnegative integers. Note the indexing, the domain starts from 1, although the range includes also 0.

A246353 gives the inverse of this sequence, in a sense that a(A246353(n)) = n for all n >= 0, and A246353(a(n)) = n for all n >= 1. When one is subtracted from the latter, another permutation of nonnegative integers is obtained: A064273. - Antti Karttunen, Aug 23 2014 based on comment from Howard A. Landman, Sep 25 2001

Also index of n-th term of A019565 when its terms are sorted in increasing order. For example: a(6) = 8. The smallest values of A019565 are 1,2,3,5,6,7 . The 6th is 7 which is A019565(8). - Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 28 2008

a(n) is the number whose binary indices are the prime indices of the n-th squarefree number (row n of A329631), where a binary index of n is any position of a 1 in its reversed binary expansion, and a prime index of n is a number m such that prime(m) divides n. The binary indices of n are row n of A048793, while the prime indices of n are row n of A112798. - Gus Wiseman, Nov 30 2019

LINKS

Table of n, a(n) for n=1..53.

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = 2^(i1-1)+2^(i2-1)+...+2^(iz-1), where A005117(n) = p_i1*p_i2*p_i3*...*p_iz.

A019565(a(n)) = A005117(n). - Peter Munn, Nov 19 2019

A000120(a(n)) = A072047(n). - Gus Wiseman, Nov 30 2019

EXAMPLE

From Gus Wiseman, Nov 30 2019: (Start)

The sequence of squarefree numbers together with their prime indices (A329631) and the number a(n) with those binary indices begins:

   1 ->  {}      ->   0

   2 ->  {1}     ->   1

   3 ->  {2}     ->   2

   5 ->  {3}     ->   4

   6 ->  {1,2}   ->   3

   7 ->  {4}     ->   8

  10 ->  {1,3}   ->   5

  11 ->  {5}     ->  16

  13 ->  {6}     ->  32

  14 ->  {1,4}   ->   9

  15 ->  {2,3}   ->   6

  17 ->  {7}     ->  64

  19 ->  {8}     -> 128

  21 ->  {2,4}   ->  10

  22 ->  {1,5}   ->  17

  23 ->  {9}     -> 256

  26 ->  {1,6}   ->  33

  29 ->  {10}    -> 512

  30 ->  {1,2,3} ->   7

(End)

MAPLE

encode_sqrfrees := proc(upto_n) local b, i; b := [ ]; for i from 1 to upto_n do if(0 <> mobius(i)) then b := [ op(b), bef(i) ]; fi; od: RETURN(b); end; # see A048623 for bef

MATHEMATICA

Join[{0}, Total[2^(PrimePi[FactorInteger[#][[All, 1]]] - 1)]& /@ Select[ Range[2, 100], SquareFreeQ]] (* Jean-Fran├žois Alcover, Mar 15 2016 *)

PROG

(PARI) lista(nn) = {for (n=1, nn, if (issquarefree(n), if (n==1, x = 0, f = factor(n); x = sum(k=1, #f~, 2^(primepi(f[k, 1])-1))); print1(x, ", "); ); ); } \\ Michel Marcus, Oct 02 2015

CROSSREFS

Inverse: A246353 (see also A064273).

Cf. A005117, A048639, A048640, A048623.

Cf. A019565.

A similar encoding of set-systems is A329661.

Cf. A000120, A048793, A056239, A070939, A072047, A112798, A329631.

Sequence in context: A322864 A054427 A232563 * A277517 A248513 A266414

Adjacent sequences:  A048669 A048670 A048671 * A048673 A048674 A048675

KEYWORD

easy,nonn

AUTHOR

Antti Karttunen, Jul 14 1999

STATUS

approved

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Last modified October 20 16:38 EDT 2020. Contains 337905 sequences. (Running on oeis4.)