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A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms. 49
1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 63, 126, 189, 378, 252, 504, 756, 1512, 315, 630, 945, 1890 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001

This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.

Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005

4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005

Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011

The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019

The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..8191 (Terms 0..1023 from T. D. Noe)

OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(0)=1; a(n+2^k)=a(n)*b(k) for n < 2^k, k = 0, 1, ... where b is A050376. - Thomas Ordowski, Mar 04 2005

The binary representation of n, n = Sum_{i=0..1+floor(log_2(n))} n_i * 2^i, n_i in {0,1}, is taken as the "Fermi-Dirac representation" (A182979) of a(n), a(n) = Product_{i=0..1+floor(log_2(n))} (b_i)^(n_i) where b_i is A050376(i), i.e., the i-th "Fermi-Dirac prime" (prime power with exponent being a power of 2). - Daniel Forgues, Feb 12 2011

From Antti Karttunen, Apr 12 & 17 2018: (Start)

a(0) = 1; a(2n) = A300841(a(n)), a(2n+1) = 2*A300841(a(n)).

a(n) = A207901(A006068(n)) = A302783(A003188(n)) = A302781(A302845(n)).

(End)

EXAMPLE

Terms following 5 are 10, 15, 30, 20, 40, 60, 120; this is followed by 7 as 6 has already occurred. - Philippe Deléham, Jun 03 2015

From Antti Karttunen, Apr 13 2018, after also Philippe Deléham's Jun 03 2015 example: (Start)

This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A300841(k), and each right hand child contains 2*A300841(k), when their parent contains k:

                                     1

                                     |

                  ...................2...................

                 3                                       6

       4......../ \........8                  12......../ \........24

      / \                 / \                 / \                 / \

     /   \               /   \               /   \               /   \

    /     \             /     \             /     \             /     \

   5       10         15       30         20       40         60      120

  7 14   21  42     28  56   84  168    35  70  105  210   140 280  420 840

  etc.

Compare also to trees like A005940 and A283477, and sequences A207901 and A302783.

(End)

MATHEMATICA

a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)

PROG

(PARI)

up_to_e = 13; \\ Good for computing up to n = (2^13)-1

v050376 = vector(up_to_e);

ispow2(n) = (n && !bitand(n, n-1));

i = 0; for(n=1, oo, if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e, break));

A050376(n) = v050376[n];

A052330(n) = { my(p=1, i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); }; \\ Antti Karttunen, Apr 12 2018

CROSSREFS

Cf. A019565, A050030, A050376 (the left edge from 2 onward), A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.

Cf. A000120, A029931, A048793, A064547, A070939, A213925, A299755, A299757, A327041.

Sequence in context: A083872 A121663 A096112 * A059900 A123664 A084980

Adjacent sequences:  A052327 A052328 A052329 * A052331 A052332 A052333

KEYWORD

nonn,look,tabf

AUTHOR

Christian G. Bower, Dec 15 1999

EXTENSIONS

Entry revised Mar 17 2005 by N. J. A. Sloane, based on comments from several people, especially David Wasserman and Thomas Ordowski

STATUS

approved

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Last modified October 23 14:43 EDT 2019. Contains 328345 sequences. (Running on oeis4.)