cp's OEIS Frontend

A336882 a(0) = 1; for k >= 0, 0 <= i < 2^k, a(2^k + i) = m_k * a(i), where m_k is the least odd number not in terms 0..2^k - 1.

%I A336882 #29 Sep 20 2020 17:02:53
%S A336882 1,3,5,15,7,21,35,105,9,27,45,135,63,189,315,945,11,33,55,165,77,231,
%T A336882 385,1155,99,297,495,1485,693,2079,3465,10395,13,39,65,195,91,273,455,
%U A336882 1365,117,351,585,1755,819,2457,4095,12285,143,429,715,2145,1001
%N A336882 a(0) = 1; for k >= 0, 0 <= i < 2^k, a(2^k + i) = m_k * a(i), where m_k is the least odd number not in terms 0..2^k - 1.
%C A336882 A permutation of the odd numbers.
%C A336882 Every positive integer, m, is the product of a unique subset of the terms of A050376. The members of the subset are often known as the Fermi-Dirac factors of m. In this sequence, the odd numbers appear lexicographically according to their Fermi-Dirac factors (with those factors listed in decreasing order). The equivalent sequence for all positive integers is A052330.
%C A336882 The sequence has a conditional exponential identity shown in the formula section. This relies on the offset being 0, as in related sequences, notably A019565 and A052330.
%H A336882 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a336/A336882.java">Java program</a> (github)
%F A336882 a(2^k) = min({ 2*m+1 : m >= 0, 2*m+1 <> a(j), 0 <= j < 2^k }) = A062090(k+2).
%F A336882 If x AND y = 0, a(x+y) = a(x) * a(y), where AND denotes the bitwise operation, A004198(.,.).
%F A336882 a(x XOR y) = A059897(a(x), a(y)), where XOR denotes bitwise exclusive-or, A003987(.,.).
%F A336882 a(x OR y) = A059896(a(x), a(y)), where OR denotes the bitwise operation, A003986(.,.).
%e A336882 a(0) = 1, as specified explicitly.
%e A336882 m_0 = 3, the least odd number not in terms 0..0.
%e A336882 So a(1) = a(2^0 + 0) = m_0 * a(0) = 3 * 1 = 3.
%e A336882 m_1 = 5, the least odd number not in terms 0..1.
%e A336882 So a(2) = a(2^1 + 0) = m_1 * a(0) = 5 * 1 = 5;
%e A336882 and a(3) = a(2^1 + 1) = m_1 * a(1) = 5 * 3 = 15.
%e A336882 The initial terms are tabulated below, equated with the product of their Fermi-Dirac factors to exhibit the lexicographic order. We start with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
%e A336882    n     a(n)
%e A336882    0    1,
%e A336882    1    3 = 3,
%e A336882    2    5 = 5,
%e A336882    3   15 = 5 * 3,
%e A336882    4    7 = 7,
%e A336882    5   21 = 7 * 3,
%e A336882    6   35 = 7 * 5,
%e A336882    7  105 = 7 * 5 * 3,
%e A336882    8    9 = 9,
%e A336882    9   27 = 9 * 3,
%e A336882   10   45 = 9 * 5,
%e A336882   11  135 = 9 * 5 * 3,
%e A336882   12   63 = 9 * 7.
%Y A336882 Permutation of A005408.
%Y A336882 Subsequence of A052330.
%Y A336882 Subsequences: A062090, A332382 (squarefree terms).
%Y A336882 A003986, A003987, A004198, A059896, A059897 are used to express relationship between terms of this sequence.
%Y A336882 Cf. A019565, A050376.
%K A336882 nonn
%O A336882 0,2
%A A336882 _Peter Munn_, Aug 16 2020