A073675 -id:A073675 - cp's OEIS frontend

A307266 Row 8 of array in A059897.

8, 4, 24, 2, 40, 12, 56, 1, 72, 20, 88, 6, 104, 28, 120, 128, 136, 36, 152, 10, 168, 44, 184, 3, 200, 52, 216, 14, 232, 60, 248, 64, 264, 68, 280, 18, 296, 76, 312, 5, 328, 84, 344, 22, 360, 92, 376, 384, 392, 100, 408, 26, 424, 108, 440, 7, 456, 116, 472, 30, 488, 124, 504, 32

Offset: 1

Peter Munn, Apr 01 2019

nonn

Also column 8 of A059897.
A self-inverse permutation of the positive integers with no fixed points; A073675 composed with A120230.
The permutation swaps pairs of integers whose ratio is 1:8 or 1:2, these ratios corresponding to the factorizations 1*8 = 2*4 = 8.

Cf. A059897, A073675, A120230, A307267.

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, 8); \\ Michel Marcus, Apr 23 2019

a(n) = A059897(8,n) = A059897(n,8).

a(n) = A073675(A120230(n)) = A120230(A073675(n)) = A073675(n) * A120230(n) / n.

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.

A366390 Dirichlet inverse of A366389.

Original entry on oeis.org

1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -26, 0, -22, 14, 15, 0, -17, 0, -25, 0, 21, 91, -29, 0, 6, 77, 0, 0, -23, -30, -31, 0, 123, 34, -28, 0, -82, 50, 75, 0, -74, -42, -106, -156, 0, 58, -122, 0, -21, -12, 51, -132, -86, 0, 142, 0, 111, 46, -110, 0, -94, 62, 0, 0, 155, -480, -97, 0, 93, 203, -113, 0, -73, 287, -66, 0, 275

Offset: 1

Antti Karttunen, Oct 22 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A010872, A030101, A057889, A073675, A366389, A366392 (rgs-transform).

Cf. also A365711.

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A073675(n) = if(valuation(n,2)%2,n/2,2*n);
A366389(n) = { my(u=A057889(n)); if(!((u-n)%3),u,A073675(u)); };
memoA366390 = Map();
A366390(n) = if(1==n,1,my(v); if(mapisdefined(memoA366390,n,&v), v, v = -sumdiv(n,d,if(dA366389(n/d)*A366390(d),0)); mapput(memoA366390,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA366389(n/d) * a(d).

A366392 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366390(i) = A366390(j) for all i, j >= 1, where A366390 is the Dirichlet inverse of A366389.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 4, 8, 9, 4, 10, 11, 12, 4, 13, 4, 14, 4, 15, 16, 17, 4, 6, 18, 4, 4, 19, 20, 21, 4, 22, 23, 24, 4, 25, 26, 27, 4, 28, 29, 30, 31, 4, 32, 33, 4, 34, 35, 36, 37, 38, 4, 39, 4, 40, 41, 42, 4, 43, 44, 4, 4, 45, 46, 47, 4, 48, 49, 50, 4, 51, 52, 53, 4, 54, 55, 56, 4, 57, 58, 59, 4, 60, 61, 15, 62

Offset: 1

Antti Karttunen, Oct 22 2023

nonn

Restricted growth sequence transform of A366390.

For all i, j: a(i) = a(j) => A365428(i) = A365428(j) => A359377(i) = A359377(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A010872, A030101, A057889, A073675, A359377, A365428, A366389, A366390.

Cf. also A366293.

\\ Needs also program given in A366389:
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA366389(n))));
A366392(n) = v366392[n];

A118966 a(n) = (n+1)/2 if n occurs among the first n-1 terms of the sequence, otherwise a(n) = 2*n - 1.

Original entry on oeis.org

1, 3, 2, 7, 9, 11, 4, 15, 5, 19, 6, 23, 25, 27, 8, 31, 33, 35, 10, 39, 41, 43, 12, 47, 13, 51, 14, 55, 57, 59, 16, 63, 17, 67, 18, 71, 73, 75, 20, 79, 21, 83, 22, 87, 89, 91, 24, 95, 97, 99, 26, 103, 105, 107, 28, 111, 29, 115, 30, 119, 121, 123, 32, 127, 129, 131, 34, 135

Offset: 1

Leroy Quet, May 07 2006

Sequence is a permutation of the positive integers. It is also its own inverse (i.e., a(a(n)) = n).
From Thomas Scheuerle, Dec 24 2020: (Start)
The same sequence can be generated by defining a(0)=0 and a(1)=1 and, for each n>1, choosing the smallest unused positive integer such that max(a(n)/n) will increase or min(a(n)/n) will decrease.
Proof: Three conditions are required to guarantee that the definitions are equivalent. The first condition is that this is a permutation; this is satisfied because this is a permutation involution. This is because (n+1)/2 is the inverse function of 2n-1, which is applied only if n is not already used in the sequence. The second condition is that, with each new term, max(a(n)/n) increases or min(a(n)/n) decreases. This is obviously the case because the next term would be either 2n-1, with would increase max(a(n)/n), or (n+1)/2, which would decrease min(a(n)/n). The third and last condition is that each new term is the smallest possible number satisfying the first two conditions. This holds because 2n-1 is the smallest possible number m*n+b where the slope m is > 1 and a(1) = 1. (A slope > 1 is needed for condition 2.)
(End)

Index entries for sequences that are permutations of the natural numbers

Cf. A118967, A073675.

% Program to test alternative definition:
%"Permutation of natural number such that max(a(n)/n)-min(a(n)/n) increases monotonously by using smallest possible next number, a(0) = 0, a(1) = 1."
function a = A118966( max_n )
    a(1) = 0;
    a(2) = 1;
    m_max = 1;
    m_min = 1;
    n = 3;
    t = 1;
    while n <= max_n
        % search next number t not yet used in a
        while ~isempty(find(a==t, 1))
            t = t+1;
        end
        m = t/(n-1);
        % check slope m
        if m < m_min || m > m_max
            % we found a candidate
            a(n) = t;
            n = n+1;
            if m > m_max
                m_max = m;
            end
            if m < m_min
                m_min = m;
            end
            t = 1;
        else
            % number t does not yet fit
            t = t+1;
        end
    end
end
% Thomas Scheuerle, Dec 24 2020

f[s_] := Block[{n = Length@s}, Append[s, If[MemberQ[s, n], (n + 1)/2, 2n - 1]]]; Rest@Nest[f, {1}, 70] (* Robert G. Wilson v, May 16 2006 *)
(* Program to test alternative definition : *)
(* "Permutation of natural number such that max(a(n)/n)-min(a(n)/n) increases monotonously by using smallest possible next number, a(0) = 0, a(1) = 1." *)
Block[{a = {0, 1}, b = {1}, c = {0}, k, r, s}, Do[k = 2; While[Nand[Set[s, Max[#] - Min[#]] > c[[-1]], FreeQ[a, k]] &@ Append[b, Set[r, k/i]], k++]; AppendTo[a, k]; AppendTo[b, r]; AppendTo[c, s], {i, 2, 55}]; a] (* Michael De Vlieger, Dec 11 2020 *)

a(n) = A073675(n-1) + 1. - Thomas Scheuerle, Dec 27 2020

More terms from Robert G. Wilson v, May 16 2006

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].

A120232 Inverse of A120231.

Original entry on oeis.org

1, 3, 2, 6, 8, 4, 5, 12, 14, 7, 17, 18, 9, 21, 10, 11, 26, 27, 13, 30, 32, 15, 35, 16, 38, 39, 41, 19, 44, 20, 47, 48, 22, 23, 53, 24, 25, 57, 59, 60, 62, 28, 29, 66, 68, 31, 71, 72, 33, 75, 34, 78, 80, 36, 37, 84, 86, 87, 89, 40, 92, 93, 42, 43, 98, 99, 45, 102, 46, 105, 107

Offset: 1

Clark Kimberling, Jun 11 2006

nonn

A permutation of the natural numbers.

			A120231, written as a function, is
{(1,1),(2,3),(3,2),(4,6),(5,7),(6,4),(7,10),(8,5)...}.
Reversing each ordered pair and reordering gives
{(1,1),(2,3),(3,2),(4,6),(5,8),(6,4),(7,5),(8,12)...}.
Suppressing first components leaves
1,3,2,6,8,4,5,12,...

Cf. A073675, A120230, A120231.

A385342 Square array A(n,k) read by descending antidiagonals, where each row n starts with an ordered list of positive integers then we swap A(n,k) with the number in the position A(n, A(n,k*n)).

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, 12, 15, 1, 15, 3, 7, 8, 14, 18, 20, 20, 2, 14, 8, 9, 16, 21, 24, 1, 24, 21, 4, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 1, 35, 2, 27, 5, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12

Offset: 1

Ali Sada, Jun 26 2025

The lower triangular array is the multiplication table.
The second row is A073675.
The first superdiagonal and the first subdiagonal are both A002378.

			The square array begins:
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
  2, 1, 6, 8, 10, 3, 14, 4, 18, 5, 22, 24, 26, 7, 30, 32, 34, ...
  3, 6, 1, 12, 15, 2, 21, 24, 27, 30, 33, 4, 39, 42, 5, 48, 51, ...
  4, 8, 12, 1, 20, 24, 28, 2, 36, 40, 44, 3, 52, 56, 60, 64, 68, ...
  5, 10, 15, 20, 1, 30, 35, 40, 45, 2, 55, 60, 65, 70, 3, 80, 85, ...
  6, 12, 18, 24, 30, 1, 42, 48, 54, 60, 66, 2, 78, 84, 90, 96, 102, ...
  7, 14, 21, 28, 35, 42, 1, 56, 63, 70, 77, 84, 91, 2, 105, 112, 119, ...
  8, 16, 24, 32, 40, 48, 56, 1, 72, 80, 88, 96, 104, 112, 120, 2, 136, ...
  9, 18, 27, 36, 45, 54, 63, 72, 1, 90, 99, 108, 117, 126, 135, 144, 153, ...
  10, 20, 30, 40, 50, 60, 70, 80, 90, 1, 110, 120, 130, 140, 150, 160, 170, ...

Cf. A002378, A073675.

cp's OEIS Frontend

A307266 Row 8 of array in A059897.

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A307267 Row 24 of array in A059897.

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A366390 Dirichlet inverse of A366389.

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A366392 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366390(i) = A366390(j) for all i, j >= 1, where A366390 is the Dirichlet inverse of A366389.

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A118966 a(n) = (n+1)/2 if n occurs among the first n-1 terms of the sequence, otherwise a(n) = 2*n - 1.

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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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A120232 Inverse of A120231.

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A385342 Square array A(n,k) read by descending antidiagonals, where each row n starts with an ordered list of positive integers then we swap A(n,k) with the number in the position A(n, A(n,k*n)).

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