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The set of Fermi-Dirac factors of A(n,k) is the union of the Fermi-Dirac factors of n and k. Symmetric square array read by antidiagonals.
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%I #24 Mar 07 2022 19:35:02

%S 1,2,2,3,2,3,4,6,6,4,5,8,3,8,5,6,10,12,12,10,6,7,6,15,4,15,6,7,8,14,6,

%T 20,20,6,14,8,9,8,21,24,5,24,21,8,9,10,18,24,28,30,30,28,24,18,10,11,

%U 10,27,8,35,6,35,8,27,10,11,12,22,30,36,40,42,42,40,36,30,22,12,13,24

%N The set of Fermi-Dirac factors of A(n,k) is the union of the Fermi-Dirac factors of n and k. Symmetric square array read by antidiagonals.

%C Every positive integer, m, is the product of a unique subset, S(m), of the numbers listed in A050376 (primes, squares of primes etc.) The Fermi-Dirac factors of m are the members of S(m). So T(n,k) is the product of the members of (S(n) U S(k)).

%C Old name: Table a(i,j) = product prime(k)^(Ei(k) OR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; OR is the bitwise operation on binary representation of the exponents.

%C Analogous to LCM, with OR replacing MAX.

%C A003418-analog seems to be A066616. - _Antti Karttunen_, Apr 12 2017

%C Considered as a binary operation, the result is the lowest common multiple of the squarefree parts of its operands multiplied by the square of the operation's result when applied to the square roots of the square parts of its operands. - _Peter Munn_, Mar 02 2022

%H Antti Karttunen, <a href="/A059896/b059896.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarePart.html">Square Part</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.

%F From _Antti Karttunen_, Apr 11 2017: (Start)

%F A(x,y) = A059895(x,y) * A059897(x,y).

%F A(x,y) * A059895(x,y) = x*y.

%F (End).

%F From _Peter Munn_, Mar 02 2022: (Start)

%F OR denotes the bitwise operation (A003986).

%F Limited multiplicative property: if gcd(n_1*k_1, n_2*k_2) = 1 then A(n_1*n_2, k_1*k_2) = A(n_1, k_1) * A(n_2, k_2).

%F For prime p, A(p^e_1, p^e_2) = p^(e_1 OR e_2).

%F A(n, A(m, k)) = A(A(n, m), k).

%F A(n, k) = A(k, n).

%F A(n, 1) = A(n, n) = n.

%F A(n^2, k^2) = A(n, k)^2.

%F A(n, k) = A(A007913(n), A007913(k)) * A(A008833(n), A008833(k)) = lcm(A007913(n), A007913(k)) * A(A000188(n), A000188(k))^2.

%F A007947(A(n, k)) = A007947(n*k).

%F Isomorphism: A(A052330(n), A052330(k)) = A052330(n OR k).

%F Equivalently, A(n, k) = A052330(A052331(n) OR A052331(k)).

%F A(A003961(n), A003961(k)) = A003961(A(n, k)).

%F A(A225546(n), A225546(k)) = A225546(A(n, k)).

%F (End)

%e A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 OR 3) * 3^(3 OR 5) = 2^7*3^7 = 279936.

%e The top left 12 X 12 corner of the array:

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

%e 2, 2, 6, 8, 10, 6, 14, 8, 18, 10, 22, 24

%e 3, 6, 3, 12, 15, 6, 21, 24, 27, 30, 33, 12

%e 4, 8, 12, 4, 20, 24, 28, 8, 36, 40, 44, 12

%e 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 55, 60

%e 6, 6, 6, 24, 30, 6, 42, 24, 54, 30, 66, 24

%e 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 77, 84

%e 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 88, 24

%e 9, 18, 27, 36, 45, 54, 63, 72, 9, 90, 99, 108

%e 10, 10, 30, 40, 10, 30, 70, 40, 90, 10, 110, 120

%e 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 11, 132

%e 12, 24, 12, 12, 60, 24, 84, 24, 108, 120, 132, 12

%t a[i_, i_] := i;

%t a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, e1[_] = 0; Scan[(e1[#[[1]]] = #[[2]])&, f1]; e2[_] = 0; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitOr[e1[#], e2[#]]& /@ Union[f1[[All, 1]], f2[[All, 1]]])];

%t Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Jun 19 2018 *)

%o (Scheme)

%o (define (A059896 n) (A059896bi (A002260 n) (A004736 n)))

%o (define (A059896bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) (* m b)) ((= 1 b) (* m a)) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A003986bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (/ a (A028233 a)) b (* m (A028233 a)))) (else (loop a (/ b (A028233 b)) (* m (A028233 b)))))))

%o ;; _Antti Karttunen_, Apr 11 2017

%o (PARI) A059896(n,k) = if(n==k, n, lcm(core(n),core(k)) * A059896(core(n,1)[2],core(k,1)[2])^2) \\ _Peter Munn_, Mar 07 2022

%Y Cf. A003418, A003990, A007947, A028233, A028234, A066616, A284576.

%Y Sequences used in a definition of this sequence: A003986, A000188/A007913/A008833, A052330/A052331.

%Y Has simple/very significant relationships with A003961, A059895/A059897, A225546, A267116.

%K base,easy,nonn,tabl

%O 1,2

%A _Marc LeBrun_, Feb 06 2001

%E New name from _Peter Munn_, Mar 02 2022