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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. 24
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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.

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)

EXAMPLE

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

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-Fran├žois Alcover, Jun 19 2018 *)

PROG

(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

(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

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.

Sequence in context: A003990 A287958 A308630 * A079542 A220370 A291048

Adjacent sequences:  A059893 A059894 A059895 * A059897 A059898 A059899

KEYWORD

base,easy,nonn,tabl

AUTHOR

Marc LeBrun, Feb 06 2001

EXTENSIONS

New name from Peter Munn, Mar 02 2022

STATUS

approved

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Last modified May 26 09:53 EDT 2022. Contains 354086 sequences. (Running on oeis4.)