A344534 -id:A344534 - cp's OEIS frontend

A344536 Inverse permutation to A344534.

0, 1, 4, 2, 32, 5, 512, 3, 16, 33, 16384, 6, 1048576, 513, 36, 8, 134217728, 17, 34359738368, 34, 516, 16385, 17592186044416, 7, 256, 1048577, 20, 514, 18014398509481984, 37, 36893488147419103232, 9, 16388, 134217729, 544, 18, 151115727451828646838272

Offset: 1

Rémy Sigrist, May 23 2021

This sequence is additive.

This sequence establishes a bijection from A225547 to A261195.

			A344534(42) = 320, so a(320) = 42.

Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A006125, A036442, A064547, A225546, A225547, A261195, A344534, A344537.

a(n) = { my (f=factor(n), v=0); for (k=1, #f~, my (x=primepi(f[k,1])-1, yy=f[k,2], y); while (yy, yy-=2^y=valuation(yy,2); v+=2^(x + (x+y)*
(x+y+1)/2))); v }

a(prime(n)) = A036442(n).
a(2^2^n) = A006125(n+1) for any n >= 0.
A000120(a(n)) = A064547(n).
a(A225546(n)) = A344537(n).
a(n) = A344537(n) iff n belongs to A225547.

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).

Original entry on oeis.org

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

Rémy Sigrist, May 22 2021

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.

			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.

Cf. A000120, A002262, A006125, A025581, A036442, A052330, A064547, A225547, A261195, A344531, A344534, A344537.

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 }

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).

cp's OEIS Frontend

A344536 Inverse permutation to A344534.

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