OFFSET
0,2
COMMENTS
The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).
The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:
...
7| 9
5| 5 8
3| 2 4 7
2| 0 1 3 6
---+--------
p/k| 0 1 2 3 ...
This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344536.
This sequence and A344535 each map between two useful choices for encoding sets of elements drawn from a 2-dimensional array. To give a very specific example, each mapping is an isomorphism between two alternative integer representations of the polynomial ring GF2[x,y]. The relevant set is {x^i*y^j : i, j >= 0}. The mappings between the two representations of the ring's addition operation are from XOR (A003987) to A059897(.,.) and for the multiplication operation, they are from A329331(.,.) to A329329(.,.). - Peter Munn, May 31 2021
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..8192
Encyclopedia of Mathematics, Isomorphism
FORMULA
a(A036442(n)) = prime(n).
a(A006125(n+1)) = 2^2^n for any n >= 0.
a(m + n) = a(m) * a(n) when m AND n = 0 (where AND denotes the bitwise AND operator).
From Peter Munn, Jun 06 2021: (Start)
(End)
EXAMPLE
For n = 42:
- 42 = 2^5 + 2^3 + 2^1,
- so we have the following Fermi-Dirac factors p^2^k:
5| X
3|
2| X X
---+------
p/k| 0 1 2
- a(42) = 2^2^1 * 2^2^2 * 5^2^0 = 320.
PROG
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, May 22 2021
STATUS
approved