A344535 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+A025581(e_k))^2^A002262(e_k) (where prime(k) denotes the k-th prime number).

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

5| 3 7

3| 1 4 8

2| 0 2 5 9

---+--------

p/k| 0 1 2 3 ...

This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344537.

This sequence establishes a bijection from A261195 to A225547.

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| X
      2|     X
    ---+------
    p/k| 0 1 2
- a(42) = 3^2^0 * 5^2^0 * 2^2^2 = 240.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• 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, A052330, A064547, A225547, A261195, A344531, A344534, A344537.

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 + A025581(e))^2^A002262(e)); v }

Formula

a(n) = A344534(A344531(n)).

a(n) = A344534(n) iff n belongs to A261195.

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

a(A006125(n)) = prime(n) for any n > 0.

a(A036442(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).