%I #14 May 24 2021 10:14:00
%S 1,2,3,6,4,8,12,24,5,10,15,30,20,40,60,120,9,18,27,54,36,72,108,216,
%T 45,90,135,270,180,360,540,1080,16,32,48,96,64,128,192,384,80,160,240,
%U 480,320,640,960,1920,144,288,432,864,576,1152,1728,3456,720,1440
%N 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).
%C The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).
%C The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:
%C ...
%C 7| 6
%C 5| 3 7
%C 3| 1 4 8
%C 2| 0 2 5 9
%C ---+--------
%C p/k| 0 1 2 3 ...
%C This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344537.
%C This sequence establishes a bijection from A261195 to A225547.
%H Rémy Sigrist, <a href="/A344535/b344535.txt">Table of n, a(n) for n = 0..8192</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(n) = A344534(A344531(n)).
%F a(n) = A344534(n) iff n belongs to A261195.
%F A064547(a(n)) = A000120(n).
%F a(A006125(n)) = prime(n) for any n > 0.
%F a(A036442(n+1)) = 2^2^n for any n >= 0.
%F a(m + n) = a(m) * a(n) when m AND n = 0 (where AND denotes the bitwise AND operator).
%e For n = 42:
%e - 42 = 2^5 + 2^3 + 2^1,
%e - so we have the following Fermi-Dirac factors p^2^k:
%e 5| X
%e 3| X
%e 2| X
%e ---+------
%e p/k| 0 1 2
%e - a(42) = 3^2^0 * 5^2^0 * 2^2^2 = 240.
%o (PARI) A002262(n)=n-binomial(round(sqrt(2+2*n)), 2)
%o A025581(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1)
%o a(n) = { my (v=1, e); while (n, n-=2^e=valuation(n, 2); v *= prime(1 + A025581(e))^2^A002262(e)); v }
%Y Cf. A000120, A002262, A006125, A025581, A036442, A052330, A064547, A225547, A261195, A344531, A344534, A344537.
%K nonn,base
%O 0,2
%A _Rémy Sigrist_, May 22 2021