A120229 -id:A120229 - cp's OEIS frontend

A120230 Split-floor-multiplier sequence (SFMS) using multipliers 1/4 and 4. (SFMS is defined at A120229.)

4, 8, 12, 1, 20, 24, 28, 2, 36, 40, 44, 3, 52, 56, 60, 64, 68, 72, 76, 5, 84, 88, 92, 6, 100, 104, 108, 7, 116, 120, 124, 128, 132, 136, 140, 9, 148, 152, 156, 10, 164, 168, 172, 11, 180, 184, 188, 192, 196, 200, 204, 13, 212, 216, 220, 14, 228, 232, 236, 15, 244, 248

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

Self-inverse permutation of the natural numbers.

			a(1)=1*4 because [1/4] is not positive.
a(2)=2*4 because [2/4] is not positive.
a(3)=3*4 because [3/4] is not positive.
a(4)=1=[4*(1/4)].
a(5)=5*4 because [5/4]=a(4), not new.

Index entries for sequences that are permutations of the natural numbers

Cf. A073675, A120229, A120230.

Row 4 and column 4 of A059897.

a(n)=[n/4] if this is positive and new, else a(n)=4n.

A120231 Split-floor-multiplier sequence (SFMS) using multipliers 2/3 and 3/2. (SFMS is defined at A120229.).

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 10, 5, 13, 15, 16, 8, 19, 9, 22, 24, 11, 12, 28, 30, 14, 33, 34, 36, 37, 17, 18, 42, 43, 20, 46, 21, 49, 51, 23, 54, 55, 25, 26, 60, 27, 63, 64, 29, 67, 69, 31, 32, 73, 75, 76, 78, 35, 81, 82, 84, 38, 87, 39, 40, 91, 41, 94, 96, 97, 44, 100, 45, 103, 105, 47

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

A permutation of the natural numbers.

			a(1)=[3/2]=1 because [2/3] is not positive.
a(2)=[2*3/2]=3.
a(3)=[3*2/3]=2.

Cf. A073675, A120230.

a(n)=[2n/3] if this is positive and new, else a(n)=[3n/2].

A120233 Split-floor-multiplier sequence (SFMS) using multipliers 3/4 and 4/3. (SFMS is defined at A120229.).

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 9, 17, 18, 11, 21, 22, 13, 25, 15, 28, 16, 30, 32, 33, 19, 20, 37, 38, 40, 23, 24, 44, 45, 26, 27, 49, 50, 29, 53, 54, 31, 57, 58, 60, 34, 35, 36, 65, 66, 68, 39, 70, 72, 41, 42, 76, 43, 78, 80, 81, 46, 47, 48, 86, 88, 89, 51, 92, 52, 94, 96

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

A permutation of the natural numbers.

Cf. A120229.

a(n)=[3n/4] if this is positive and new, else a(n)=[4n/3].

A120235 Split-floor-multiplier sequence (SFMS) using multipliers 2/5 and 5/2. (SFMS is defined at A120229.).

Original entry on oeis.org

2, 5, 1, 10, 12, 15, 17, 3, 22, 4, 27, 30, 32, 35, 6, 40, 42, 7, 47, 8, 52, 55, 9, 60, 62, 65, 67, 11, 72, 75, 77, 80, 13, 85, 14, 90, 92, 95, 97, 16, 102, 105, 107, 110, 18, 115, 117, 19, 122, 20, 127, 130, 21, 135, 140, 142, 23, 147, 24, 152, 155, 25, 160, 26, 165, 167

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

A permutation of the natural numbers.

Cf. A120229, A120236.

a(n)=[2n/5] if this is positive and new, else a(n)=[5n/2].

A120237 Split-floor-multiplier sequence (SFMS) using multipliers 2^(-1/2) and 2^(1/2). (SFMS is defined at A120229.).

Original entry on oeis.org

1, 2, 4, 5, 3, 8, 9, 11, 6, 7, 15, 16, 18, 19, 10, 22, 12, 25, 13, 14, 29, 31, 32, 33, 17, 36, 38, 39, 20, 21, 43, 45, 23, 24, 49, 50, 26, 53, 27, 28, 57, 59, 30, 62, 63, 65, 66, 67, 34, 35, 72, 73, 37, 76, 77, 79, 40, 41, 83, 42, 86, 87, 44, 90, 91, 46, 47, 48, 97, 98, 100, 101

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

A permutation of the natural numbers.

Cf. A120229, A120238.

a(n)=[nr] if this is positive and new, else a(n)=[n/r], where r=2^(-1/2).

A120239 Split-floor-multiplier sequence (SFMS) using multipliers 1/tau and tau, where tau=(1+sqrt(5))/2. (SFMS is defined at A120229.).

Original entry on oeis.org

1, 3, 4, 2, 8, 9, 11, 12, 5, 6, 17, 7, 21, 22, 24, 25, 10, 29, 30, 32, 33, 13, 14, 38, 15, 16, 43, 45, 46, 18, 19, 51, 20, 55, 56, 58, 59, 23, 63, 64, 66, 67, 26, 27, 72, 28, 76, 77, 79, 80, 31, 84, 85, 87, 88, 34, 35, 93, 36, 37, 98, 100, 101, 39, 40, 106, 41, 42, 111, 113

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

A permutation of the natural numbers.

Cf. A120229, A120240.

a(n)=[nr] if this is positive and new, else a(n)=[n/r], where r=2/(1+sqrt(5)).

A059897 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.

Original entry on oeis.org

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

Offset: 1

Marc LeBrun, Feb 06 2001

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.
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 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).
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.
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)
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

			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 integers > 1 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)

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

Cf. A000040, A003987, A003991, A028233, A028234, A050376, A059896, A089913, A207901, A268387, A284577, A302033.
Cf. A284567 (A000142 or A003418-analog for this operation).
Rows/columns: A073675 (2), A120229 (3), A120230 (4), A307151 (5), A307150 (6), A307266 (8), A307267 (24).
Particularly significant subgroups or cosets: A000028, A000379, A003159, A005117, A030229, A252895. See also the lists in A329050, A352273.
Sequences that relate this sequence to multiplication: A000188, A007913, A059895.

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-François Alcover, Jun 19 2018 *)

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

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

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

(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

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(n*m,m) = n, A(n*m,k) = A(n,k) * A(m,k) / k. - Peter Munn, Apr 04 2019
A(n,k) = A007913(n*k) * A(A000188(n), A000188(k))^2. - Peter Munn, Mar 21 2022

New name from Peter Munn, Mar 21 2022

A307150 Row 6 of array in A059897.

Original entry on oeis.org

6, 3, 2, 24, 30, 1, 42, 12, 54, 15, 66, 8, 78, 21, 10, 96, 102, 27, 114, 120, 14, 33, 138, 4, 150, 39, 18, 168, 174, 5, 186, 48, 22, 51, 210, 216, 222, 57, 26, 60, 246, 7, 258, 264, 270, 69, 282, 32, 294, 75, 34, 312, 318, 9, 330, 84, 38, 87, 354, 40, 366, 93

Offset: 1

N. J. A. Sloane, Mar 29 2019

nonn

From Peter Munn, Apr 02 2019: (Start)
Also column 6 of A059897.
A self-inverse permutation of the positive integers with no fixed points; A073675 composed with A120229.
The permutation swaps pairs of integers whose ratio is 1:6 or 2:3, these ratios corresponding to the factorizations 1*6 = 2*3 = 6. Row 6 is the first row of A059897 to exhibit more than 1 such ratio.
(End)
The integers in the pairs with ratio 1:6 are listed in A036668, the integers in the pairs with ratio 2:3 are listed in A325424. - Peter Munn, Mar 05 2020

Cf. A036668, A059897, A073675, A120229, A325424.

From Peter Munn, Apr 02 2019: (Start)
a(n) = A059897(6,n) = A059897(n,6).
a(n) = A073675(A120229(n)) = A120229(A073675(n)) = A073675(n) * A120229(n) / n.
(End)

More terms from Alois P. Heinz, Mar 31 2019

A307267 Row 24 of array in A059897.

Original entry on oeis.org

24, 12, 8, 6, 120, 4, 168, 3, 216, 60, 264, 2, 312, 84, 40, 384, 408, 108, 456, 30, 56, 132, 552, 1, 600, 156, 72, 42, 696, 20, 744, 192, 88, 204, 840, 54, 888, 228, 104, 15, 984, 28, 1032, 66, 1080, 276, 1128, 128, 1176, 300, 136, 78, 1272, 36, 1320, 21, 152, 348, 1416, 10

Offset: 1

Peter Munn, Apr 01 2019

nonn

Also column 24 of A059897.
A self-inverse permutation of the positive integers with no fixed points; a composition of A073675, A120229 and A120230.
The permutation swaps pairs of integers whose ratio is 1:24, 1:6, 3:8 or 2:3, these ratios corresponding to the factorizations 1*24 = 2*12 = 3*8 = 4*6 = 24. Row 24 is the first row of A059897 to exhibit more than 2 such ratios.

Cf. A059897, A073675, A120229, A120230, A307266.

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])))); }
a(n) = T(n, 24); \\ Michel Marcus, Apr 23 2019

a(n) = A059897(24,n) = A059897(n,24).

a(n) = A073675(A120229(A120230(n))) = A073675(n) * A120229(n) * A120230(n) / n^2.

A120236 Inverse of A120235.

Original entry on oeis.org

3, 1, 8, 10, 2, 15, 18, 20, 23, 4, 28, 5, 33, 35, 6, 40, 7, 45, 48, 50, 53, 9, 58, 60, 63, 65, 11, 70, 73, 12, 78, 13, 83, 85, 14, 90, 93, 95, 98, 16, 103, 17, 108, 110, 113, 115, 19, 120, 123, 125, 128, 21, 133, 135, 22, 140, 143, 145, 148, 24, 153, 25, 138, 160, 26, 165

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

A permutation of the natural numbers.

Cf. A120229, A120235.

cp's OEIS Frontend

A120230 Split-floor-multiplier sequence (SFMS) using multipliers 1/4 and 4. (SFMS is defined at A120229.)

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A120231 Split-floor-multiplier sequence (SFMS) using multipliers 2/3 and 3/2. (SFMS is defined at A120229.).

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A120233 Split-floor-multiplier sequence (SFMS) using multipliers 3/4 and 4/3. (SFMS is defined at A120229.).

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A120235 Split-floor-multiplier sequence (SFMS) using multipliers 2/5 and 5/2. (SFMS is defined at A120229.).

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A120237 Split-floor-multiplier sequence (SFMS) using multipliers 2^(-1/2) and 2^(1/2). (SFMS is defined at A120229.).

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A120239 Split-floor-multiplier sequence (SFMS) using multipliers 1/tau and tau, where tau=(1+sqrt(5))/2. (SFMS is defined at A120229.).

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A059897 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.

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A307150 Row 6 of array in A059897.

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A307267 Row 24 of array in A059897.

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