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%I #95 Apr 16 2024 13:54:59
%S 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,
%T 70,105,210,140,280,420,840,9,18,27,54,36,72,108,216,45,90,135,270,
%U 180,360,540,1080,63,126,189,378,252,504,756,1512,315,630,945,1890
%N 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.
%C Inverse of sequence A064358 considered as a permutation of the positive integers. - _Howard A. Landman_, Sep 25 2001
%C 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.
%C Are there any other values of n besides 4 and 36 with a(n) = n? - _Thomas Ordowski_, Apr 01 2005
%C 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
%C 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
%C The subsequence consisting of the squarefree terms is A019565. - _Peter Munn_, Mar 28 2018
%C 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
%C 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
%H Antti Karttunen, <a href="/A052330/b052330.txt">Table of n, a(n) for n = 0..8191</a> (Terms 0..1023 from T. D. Noe)
%H Michael De Vlieger, <a href="/A052330/a052330.png">Fan style binary tree showing a(n)</a>, n = 0..16383, with a color function representing primes in red, perfect powers of primes in gold, squarefree composites in greens, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers. This is a 15 level version of Karttunen's diagram in the example.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Ordering_of_positive_integers_by_increasing_%22Fermi-Dirac_representation%22">Ordering of positive integers by increasing "Fermi-Dirac representation"</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F 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
%F 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
%F From _Antti Karttunen_, Apr 12 & 17 2018: (Start)
%F a(0) = 1; a(2n) = A300841(a(n)), a(2n+1) = 2*A300841(a(n)).
%F a(n) = A207901(A006068(n)) = A302783(A003188(n)) = A302781(A302845(n)).
%F (End)
%e 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
%e From _Antti Karttunen_, Apr 13 2018, after also _Philippe Deléham_'s Jun 03 2015 example: (Start)
%e 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:
%e 1
%e |
%e ...................2...................
%e 3 6
%e 4......../ \........8 12......../ \........24
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e 5 10 15 30 20 40 60 120
%e 7 14 21 42 28 56 84 168 35 70 105 210 140 280 420 840
%e etc.
%e Compare also to trees like A005940 and A283477, and sequences A207901 and A302783.
%e (End)
%t a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* _Ivan Neretin_, May 09 2015 *)
%o (PARI)
%o up_to_e = 13; \\ Good for computing up to n = (2^13)-1
%o v050376 = vector(up_to_e);
%o ispow2(n) = (n && !bitand(n,n-1));
%o i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
%o A050376(n) = v050376[n];
%o 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
%Y Subsequences: A019565 (squarefree terms), A050376 (the left edge from 2 onward), A336882 (odd terms).
%Y Cf. A050030, A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.
%Y Cf. A000120, A029931, A048793, A064547, A070939, A213925, A299755, A299757, A327041.
%K nonn,look,tabf,changed
%O 0,2
%A _Christian G. Bower_, Dec 15 1999
%E Entry revised Mar 17 2005 by _N. J. A. Sloane_, based on comments from several people, especially _David Wasserman_ and _Thomas Ordowski_
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