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%I #47 Mar 24 2022 15:35:49
%S 1,2,2,3,1,3,4,6,6,4,5,8,1,8,5,6,10,12,12,10,6,7,3,15,1,15,3,7,8,14,2,
%T 20,20,2,14,8,9,4,21,24,1,24,21,4,9,10,18,24,28,30,30,28,24,18,10,11,
%U 5,27,2,35,1,35,2,27,5,11,12,22,30,36,40,42,42,40,36,30,22,12,13,24,33
%N Symmetric square array read by antidiagonals: A(n,k) is the product of all factors that occur in one, but not both, of the Fermi-Dirac factorizations of n and k.
%C Old name: Square array read by antidiagonals: T(i,j) = product prime(k)^(Ei(k) XOR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; XOR is the bitwise operation on binary representation of the exponents.
%C Analogous to multiplication, with XOR replacing +.
%C From _Peter Munn_, Apr 01 2019: (Start)
%C (1) Defines an abelian group whose underlying set is the positive integers. (2) Every element is self-inverse. (3) For all n and k, A(n,k) is a divisor of n*k. (4) The terms of A050376, sometimes called Fermi-Dirac primes, form a minimal set of generators. In ordered form, it is the lexicographically earliest such set.
%C The unique factorization of positive integers into products of distinct terms of the group's lexicographically earliest minimal set of generators seems to follow from (1) (2) and (3).
%C From (1) and (2), every row and every column of the table is a self-inverse permutation of the positive integers. Rows/columns numbered by nonmembers of A050376 are compositions of earlier rows/columns.
%C It is a subgroup of the equivalent group over the nonzero integers, which has -1 as an additional generator.
%C As generated by A050376, the subgroup of even length words is A000379. The complementary set of odd length words is A000028.
%C The subgroup generated by A000040 (the primes) is A005117 (the squarefree numbers).
%C (End)
%C Considered as a binary operation, the result is (the squarefree part of the product of its operands) times the square of (the operation's result when applied to the square roots of the square parts of its operands). - _Peter Munn_, Mar 21 2022
%H Antti Karttunen, <a href="/A059897/b059897.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="http://mathworld.wolfram.com/Group.html">Group</a>, <a href="http://mathworld.wolfram.com/SquarePart.html">Square Part</a>, <a href="http://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.
%F For all x, y >= 1, A(x,y) * A059895(x,y)^2 = x*y. - _Antti Karttunen_, Apr 11 2017
%F From _Peter Munn_, Apr 01 2019: (Start)
%F A(n,1) = A(1,n) = n
%F A(n, A(m,k)) = A(A(n,m), k)
%F A(n,n) = 1
%F A(n,k) = A(k,n)
%F if i_1 <> i_2 then A(A050376(i_1), A050376(i_2)) = A050376(i_1) * A050376(i_2)
%F if A(n,k_1) = n * k_1 and A(n,k_2) = n * k_2 then A(n, A(k_1,k_2)) = n * A(k_1,k_2)
%F (End)
%F T(k, m) = k*m for coprime k and m. - _David A. Corneth_, Apr 03 2019
%F if A(n*m,m) = n, A(n*m,k) = A(n,k) * A(m,k) / k. - _Peter Munn_, Apr 04 2019
%F A(n,k) = A007913(n*k) * A(A000188(n), A000188(k))^2. - _Peter Munn_, Mar 21 2022
%e A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 XOR 3) * 3^(3 XOR 5) = 2^6 * 3^6 = 46656.
%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, 1, 6, 8, 10, 3, 14, 4, 18, 5, 22, 24
%e 3, 6, 1, 12, 15, 2, 21, 24, 27, 30, 33, 4
%e 4, 8, 12, 1, 20, 24, 28, 2, 36, 40, 44, 3
%e 5, 10, 15, 20, 1, 30, 35, 40, 45, 2, 55, 60
%e 6, 3, 2, 24, 30, 1, 42, 12, 54, 15, 66, 8
%e 7, 14, 21, 28, 35, 42, 1, 56, 63, 70, 77, 84
%e 8, 4, 24, 2, 40, 12, 56, 1, 72, 20, 88, 6
%e 9, 18, 27, 36, 45, 54, 63, 72, 1, 90, 99, 108
%e 10, 5, 30, 40, 2, 15, 70, 20, 90, 1, 110, 120
%e 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1, 132
%e 12, 24, 4, 3, 60, 8, 84, 6, 108, 120, 132, 1
%e From _Peter Munn_, Apr 04 2019: (Start)
%e The subgroup generated by {6,8,10}, the first three integers > 1 not in A050376, has the following table:
%e 1 6 8 10 12 15 20 120
%e 6 1 12 15 8 10 120 20
%e 8 12 1 20 6 120 10 15
%e 10 15 20 1 120 6 8 12
%e 12 8 6 120 1 20 15 10
%e 15 10 120 6 20 1 12 8
%e 20 120 10 8 15 12 1 6
%e 120 20 15 12 10 8 6 1
%e (End)
%t a[i_, i_] = 1;
%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 @@ (#^BitXor[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 (A059897 n) (A059897bi (A002260 n) (A004736 n)))
%o (define (A059897bi 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) (A003987bi (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) T(n,k) = {if (n==1, return (k)); if (k==1, return (n)); my(fn=factor(n), fk=factor(k)); vp = setunion(fn[,1]~, fk[,1]~); prod(i=1, #vp, vp[i]^(bitxor(valuation(n, vp[i]), valuation(k, vp[i]))));} \\ _Michel Marcus_, Apr 03 2019
%o (PARI) T(i, j) = {if(gcd(i, j) == 1, return(i * j)); if(i == j, return(1)); my(f = vecsort(concat(factor(i)~, factor(j)~)), t = 1, res = 1); while(t + 1 <= #f, if(f[1, t] == f[1, t+1], res *= f[1, t] ^ bitxor(f[2, t] , f[2, t+1]); t+=2; , res*= f[1, t]^f[2, t]; t++; ) ); if(t == #f, res *= f[1, #f] ^ f[2, #f]); res } \\ _David A. Corneth_, Apr 03 2019
%o (PARI) A059897(n,k) = if(n==k, 1, core(n*k) * A059897(core(n,1)[2],core(k,1)[2])^2) \\ _Peter Munn_, Mar 21 2022
%Y Cf. A000040, A003987, A003991, A028233, A028234, A050376, A059896, A089913, A207901, A268387, A284577, A302033.
%Y Cf. A284567 (A000142 or A003418-analog for this operation).
%Y Rows/columns: A073675 (2), A120229 (3), A120230 (4), A307151 (5), A307150 (6), A307266 (8), A307267 (24).
%Y Particularly significant subgroups or cosets: A000028, A000379, A003159, A005117, A030229, A252895. See also the lists in A329050, A352273.
%Y Sequences that relate this sequence to multiplication: A000188, A007913, A059895.
%K base,easy,nonn,tabl,nice,look
%O 1,2
%A _Marc LeBrun_, Feb 06 2001
%E New name from _Peter Munn_, Mar 21 2022
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