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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).
3
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
0,2
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| 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.
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).
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| X
2| X
---+------
p/k| 0 1 2
- a(42) = 3^2^0 * 5^2^0 * 2^2^2 = 240.
PROG
(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 }
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, May 22 2021
STATUS
approved