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)
Michael De Vlieger, Fan style binary tree showing a(n), 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.
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)
(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.
(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
KEYWORD
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