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A059897 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. 31
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Analogous to multiplication, with XOR replacing +.

From Peter Munn, Apr 01 2019: (Start)

(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.

The unique factorisation 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).

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 non-members of A050376 are compositions of earlier rows/columns.

It is a subgroup of the equivalent group over the nonzero integers, which has -1 as an additional generator.

As generated by A050376, the subgroup of even length words is A000379. The complementary set of odd length words is A000028.

The subgroup generated by A000040 (the primes) is A005117 (the squarefree numbers).

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

FORMULA

For all x, y >= 1, A(x,y) * A059895(x,y)^2 = x*y. - Antti Karttunen, Apr 11 2017

From Peter Munn, Apr 01 2019: (Start)

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

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

A(n,n) = 1

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

if i_1 <> i_2 then A(A050376(i_1), A050376(i_2)) = A050376(i_1) * A050376(i_2)

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)

(End)

T(k, m) = k*m for coprime k and m. - David A. Corneth, Apr 03 2019

if A(nm,m) = n, A(nm,k) = A(n,k) * A(m,k) / k. - Peter Munn, Apr 04 2019

EXAMPLE

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.

The top left 12 X 12 corner of the array:

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

   2,  1,  6,  8, 10,  3, 14,  4,  18,   5,  22,  24

   3,  6,  1, 12, 15,  2, 21, 24,  27,  30,  33,   4

   4,  8, 12,  1, 20, 24, 28,  2,  36,  40,  44,   3

   5, 10, 15, 20,  1, 30, 35, 40,  45,   2,  55,  60

   6,  3,  2, 24, 30,  1, 42, 12,  54,  15,  66,   8

   7, 14, 21, 28, 35, 42,  1, 56,  63,  70,  77,  84

   8,  4, 24,  2, 40, 12, 56,  1,  72,  20,  88,   6

   9, 18, 27, 36, 45, 54, 63, 72,   1,  90,  99, 108

  10,  5, 30, 40,  2, 15, 70, 20,  90,   1, 110, 120

  11, 22, 33, 44, 55, 66, 77, 88,  99, 110,   1, 132

  12, 24,  4,  3, 60,  8, 84,  6, 108, 120, 132,   1

From Peter Munn, Apr 04 2019: (Start)

The subgroup generated by {6,8,10}, the first three positive integers not in A050376, has the following table:

    1     6     8    10    12    15    20   120

    6     1    12    15     8    10   120    20

    8    12     1    20     6   120    10    15

   10    15    20     1   120     6     8    12

   12     8     6   120     1    20    15    10

   15    10   120     6    20     1    12     8

   20   120    10     8    15    12     1     6

  120    20    15    12    10     8     6     1

(End)

MATHEMATICA

a[i_, i_] = 1;

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]]])];

Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-Fran├žois Alcover, Jun 19 2018 *)

PROG

(Scheme)

(define (A059897 n) (A059897bi (A002260 n) (A004736 n)))

(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)))))))

;; Antti Karttunen, Apr 11 2017

(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

(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

CROSSREFS

Cf. A000028, A000040, A000379, A003987, A003991, A005117, A028233, A028234, A050376, A059895, A059896, A089913, A207901, A268387, A284577, A302033.

Cf. A284567 (A000142 or A003418-analog for this operation).

Rows 2 through 6 are A073675, A120229, A120230, A307151, A307150.

Sequence in context: A280172 A089913 A257522 * A325821 A303719 A299595

Adjacent sequences:  A059894 A059895 A059896 * A059898 A059899 A059900

KEYWORD

base,easy,nonn,tabl,nice,look

AUTHOR

Marc LeBrun, Feb 06 2001

STATUS

approved

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Last modified December 11 07:29 EST 2019. Contains 329914 sequences. (Running on oeis4.)