A059896 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.

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

Offset: 1

Author

Marc LeBrun, Feb 06 2001

Keywords

• base
• easy
• nonn
• tabl

Comments

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)).

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.

Analogous to LCM, with OR replacing MAX.

A003418-analog seems to be A066616. - Antti Karttunen, Apr 12 2017

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

Examples

			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.
The top left 12 X 12 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12
   2,  2,  6,  8, 10,  6, 14,  8,  18,  10,  22,  24
   3,  6,  3, 12, 15,  6, 21, 24,  27,  30,  33,  12
   4,  8, 12,  4, 20, 24, 28,  8,  36,  40,  44,  12
   5, 10, 15, 20,  5, 30, 35, 40,  45,  10,  55,  60
   6,  6,  6, 24, 30,  6, 42, 24,  54,  30,  66,  24
   7, 14, 21, 28, 35, 42,  7, 56,  63,  70,  77,  84
   8,  8, 24,  8, 40, 24, 56,  8,  72,  40,  88,  24
   9, 18, 27, 36, 45, 54, 63, 72,   9,  90,  99, 108
  10, 10, 30, 40, 10, 30, 70, 40,  90,  10, 110, 120
  11, 22, 33, 44, 55, 66, 77, 88,  99, 110,  11, 132
  12, 24, 12, 12, 60, 24, 84, 24, 108, 120, 132,  12
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array
• Eric Weisstein's World of Mathematics, Square Part.
• Eric Weisstein's World of Mathematics, Squarefree Part.

Crossrefs

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

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

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

Programs

Mathematica

a[i_, i_] := i;
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]]])];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

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

Scheme

(define (A059896 n) (A059896bi (A002260 n) (A004736 n)))
(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)))))))
;; Antti Karttunen, Apr 11 2017

Formula

From Antti Karttunen, Apr 11 2017: (Start)

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

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

(End).

From Peter Munn, Mar 02 2022: (Start)

OR denotes the bitwise operation (A003986).

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).

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

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

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

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

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

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

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

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

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

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

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

(End)

Extensions

New name from Peter Munn, Mar 02 2022