cp's OEIS Frontend

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).

%I A344534 #20 Jun 25 2021 04:08:42
%S A344534 1,2,4,8,3,6,12,24,16,32,64,128,48,96,192,384,9,18,36,72,27,54,108,
%T A344534 216,144,288,576,1152,432,864,1728,3456,5,10,20,40,15,30,60,120,80,
%U A344534 160,320,640,240,480,960,1920,45,90,180,360,135,270,540,1080,720,1440
%N 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).
%C A344534 The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n).
%C A344534 The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:
%C A344534     ...
%C A344534       7| 9
%C A344534       5| 5 8
%C A344534       3| 2 4 7
%C A344534       2| 0 1 3 6
%C A344534     ---+--------
%C A344534     p/k| 0 1 2 3 ...
%C A344534 This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344536.
%C A344534 This sequence establishes a bijection from A261195 to A225547.
%C A344534 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
%H A344534 Rémy Sigrist, <a href="/A344534/b344534.txt">Table of n, a(n) for n = 0..8192</a>
%H A344534 Encyclopedia of Mathematics, <a href="https://encyclopediaofmath.org/wiki/Isomorphism">Isomorphism</a>
%H A344534 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A344534 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A344534 a(n) = A344535(A344531(n)).
%F A344534 a(n) = A344535(n) iff n belongs to A261195.
%F A344534 A064547(a(n)) = A000120(n).
%F A344534 a(A036442(n)) = prime(n).
%F A344534 a(A006125(n+1)) = 2^2^n for any n >= 0.
%F A344534 a(m + n) = a(m) * a(n) when m AND n = 0 (where AND denotes the bitwise AND operator).
%F A344534 From _Peter Munn_, Jun 06 2021: (Start)
%F A344534 a(n) = A225546(A344535(n)).
%F A344534 a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
%F A344534 a(A329331(n, k)) = A329329(a(n), a(k)).
%F A344534 (End)
%e A344534 For n = 42:
%e A344534 - 42 = 2^5 + 2^3 + 2^1,
%e A344534 - so we have the following Fermi-Dirac factors p^2^k:
%e A344534       5| X
%e A344534       3|
%e A344534       2|   X X
%e A344534     ---+------
%e A344534     p/k| 0 1 2
%e A344534 - a(42) = 2^2^1 * 2^2^2 * 5^2^0 = 320.
%o A344534 (PARI) A002262(n)=n-binomial(round(sqrt(2+2*n)), 2)
%o A344534 A025581(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1)
%o A344534 a(n) = { my (v=1, e); while (n, n-=2^e=valuation(n, 2); v* = prime(1 + A002262(e))^2^A025581(e)); v }
%Y A344534 Cf. A000120, A002262, A006125, A025581, A036442, A064547, A225546, A225547, A261195, A329050, A344531, A344535, A344536.
%Y A344534 Comparable mappings that also use Fermi-Dirac factors: A052330, A059900.
%Y A344534 Maps binary operations A003987 to A059897, A329331 to A329329.
%K A344534 nonn,base
%O A344534 0,2
%A A344534 _Rémy Sigrist_, May 22 2021