A344534 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+A002262(e_k))^2^A025581(e_k) (where prime(k) denotes the k-th prime number).

1, 2, 4, 8, 3, 6, 12, 24, 16, 32, 64, 128, 48, 96, 192, 384, 9, 18, 36, 72, 27, 54, 108, 216, 144, 288, 576, 1152, 432, 864, 1728, 3456, 5, 10, 20, 40, 15, 30, 60, 120, 80, 160, 320, 640, 240, 480, 960, 1920, 45, 90, 180, 360, 135, 270, 540, 1080, 720, 1440

Offset: 0

Author

Rémy Sigrist, May 22 2021

Keywords

• nonn
• base

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 establishes a bijection from A261195 to A225547.

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

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Encyclopedia of Mathematics, Isomorphism
• Index entries for sequences related to binary expansion of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000120, A002262, A006125, A025581, A036442, A064547, A225546, A225547, A261195, A329050, A344531, A344535, A344536.

Comparable mappings that also use Fermi-Dirac factors: A052330, A059900.

Maps binary operations A003987 to A059897, A329331 to A329329.

Programs

PARI

A002262(n)=n-binomial(round(sqrt(2+2*n)), 2)
A025581(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1)
a(n) = { my (v=1, e); while (n, n-=2^e=valuation(n, 2); v* = prime(1 + A002262(e))^2^A025581(e)); v }

Formula

a(n) = A344535(A344531(n)).

a(n) = A344535(n) iff n belongs to A261195.

A064547(a(n)) = A000120(n).

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)

a(n) = A225546(A344535(n)).

a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.

a(A329331(n, k)) = A329329(a(n), a(k)).

(End)