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%I #29 Sep 20 2020 17:02:53
%S 1,3,5,15,7,21,35,105,9,27,45,135,63,189,315,945,11,33,55,165,77,231,
%T 385,1155,99,297,495,1485,693,2079,3465,10395,13,39,65,195,91,273,455,
%U 1365,117,351,585,1755,819,2457,4095,12285,143,429,715,2145,1001
%N 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 A permutation of the odd numbers.
%C 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 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 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a336/A336882.java">Java program</a> (github)
%F a(2^k) = min({ 2*m+1 : m >= 0, 2*m+1 <> a(j), 0 <= j < 2^k }) = A062090(k+2).
%F If x AND y = 0, a(x+y) = a(x) * a(y), where AND denotes the bitwise operation, A004198(.,.).
%F a(x XOR y) = A059897(a(x), a(y)), where XOR denotes bitwise exclusive-or, A003987(.,.).
%F a(x OR y) = A059896(a(x), a(y)), where OR denotes the bitwise operation, A003986(.,.).
%e a(0) = 1, as specified explicitly.
%e m_0 = 3, the least odd number not in terms 0..0.
%e So a(1) = a(2^0 + 0) = m_0 * a(0) = 3 * 1 = 3.
%e m_1 = 5, the least odd number not in terms 0..1.
%e So a(2) = a(2^1 + 0) = m_1 * a(0) = 5 * 1 = 5;
%e and a(3) = a(2^1 + 1) = m_1 * a(1) = 5 * 3 = 15.
%e 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 n a(n)
%e 0 1,
%e 1 3 = 3,
%e 2 5 = 5,
%e 3 15 = 5 * 3,
%e 4 7 = 7,
%e 5 21 = 7 * 3,
%e 6 35 = 7 * 5,
%e 7 105 = 7 * 5 * 3,
%e 8 9 = 9,
%e 9 27 = 9 * 3,
%e 10 45 = 9 * 5,
%e 11 135 = 9 * 5 * 3,
%e 12 63 = 9 * 7.
%Y Permutation of A005408.
%Y Subsequence of A052330.
%Y Subsequences: A062090, A332382 (squarefree terms).
%Y A003986, A003987, A004198, A059896, A059897 are used to express relationship between terms of this sequence.
%Y Cf. A019565, A050376.
%K nonn
%O 0,2
%A _Peter Munn_, Aug 16 2020
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