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.

1, 3, 5, 15, 7, 21, 35, 105, 9, 27, 45, 135, 63, 189, 315, 945, 11, 33, 55, 165, 77, 231, 385, 1155, 99, 297, 495, 1485, 693, 2079, 3465, 10395, 13, 39, 65, 195, 91, 273, 455, 1365, 117, 351, 585, 1755, 819, 2457, 4095, 12285, 143, 429, 715, 2145, 1001

Offset

0,2

Comments

A permutation of the odd numbers.

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.

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.

Links

Table of n, a(n) for n=0..52.

Sean A. Irvine, Java program (github)

Formula

a(2^k) = min({ 2*m+1 : m >= 0, 2*m+1 <> a(j), 0 <= j < 2^k }) = A062090(k+2).

If x AND y = 0, a(x+y) = a(x) * a(y), where AND denotes the bitwise operation, A004198(.,.).

a(x XOR y) = A059897(a(x), a(y)), where XOR denotes bitwise exclusive-or, A003987(.,.).

a(x OR y) = A059896(a(x), a(y)), where OR denotes the bitwise operation, A003986(.,.).

Example

a(0) = 1, as specified explicitly.
m_0 = 3, the least odd number not in terms 0..0.
So a(1) = a(2^0 + 0) = m_0 * a(0) = 3 * 1 = 3.
m_1 = 5, the least odd number not in terms 0..1.
So a(2) = a(2^1 + 0) = m_1 * a(0) = 5 * 1 = 5;
and a(3) = a(2^1 + 1) = m_1 * a(1) = 5 * 3 = 15.
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.
   n     a(n)
   0    1,
   1    3 = 3,
   2    5 = 5,
   3   15 = 5 * 3,
   4    7 = 7,
   5   21 = 7 * 3,
   6   35 = 7 * 5,
   7  105 = 7 * 5 * 3,
   8    9 = 9,
   9   27 = 9 * 3,
  10   45 = 9 * 5,
  11  135 = 9 * 5 * 3,
  12   63 = 9 * 7.

Crossrefs

Permutation of A005408.

Subsequence of A052330.

Subsequences: A062090, A332382 (squarefree terms).

A003986, A003987, A004198, A059896, A059897 are used to express relationship between terms of this sequence.

Cf. A019565, A050376.

Sequence in context: A058220 A100181 A180620 * A332382 A277323 A340194

Adjacent sequences: A336879 A336880 A336881 * A336883 A336884 A336885

Keyword

nonn

Author

Peter Munn, Aug 16 2020

Status

approved