A323881 -id:A323881 - cp's OEIS frontend

A353461 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A323881 (the Dirichlet inverse of A126760).

1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 3, 0, 3, 0, 3, 0, 4, 0, 4, 0, 2, 0, 9, 0, 5, 0, 5, 0, 7, 0, 1, 0, 6, 0, 8, 0, 7, 0, 7, 0, 9, 0, 8, 0, 5, 0, 11, 0, 9, 0, 1, 0, 12, 0, 10, 0, 10, 0, 12, 0, 2, 0, 11, 0, 15, 0, 12, 0, 12, 0, 10, 0, 3, 0, 13, 0, 27, 0, 14, 0, 2, 0, 19, 0, 15, 0, 4, 0, 20, 0, 3, 0, 16, 0, 21

Offset: 1

Antti Karttunen, Apr 20 2022

sign

Taking the Dirichlet convolution between this sequence and A349393 gives A349371, and similarly for many other such analogous pairs.

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

Cf. A003602, A126760, A323881, A353460 (Dirichlet inverse), A353462 (sum with it).

Cf. also A349371, A349393.

up_to = 65537;
A003602(n) = (1+(n>>valuation(n,2)))/2;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA126760(n)));
A323881(n) = v323881[n];
A353461(n) = sumdiv(n,d,A003602(d)*A323881(n/d));

a(n) = Sum_{d|n} A003602(d) * A323881(n/d).

a(n) = A353462(n) - A353460(n).

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 5, 3, 2, 1, 6, 1, 7, 2, 3, 4, 8, 1, 9, 5, 1, 3, 10, 2, 11, 1, 4, 6, 12, 1, 13, 7, 5, 2, 14, 3, 15, 4, 2, 8, 16, 1, 17, 9, 6, 5, 18, 1, 19, 3, 7, 10, 20, 2, 21, 11, 3, 1, 22, 4, 23, 6, 8, 12, 24, 1, 25, 13, 9, 7, 26, 5, 27, 2, 1, 14, 28, 3, 29, 15, 10, 4, 30, 2

Offset: 0

N. J. A. Sloane, Feb 19 2007

nonn

For further information see A126759, which provided the original motivation for this sequence.
From Antti Karttunen, Jan 28 2015: (Start)
The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.
A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).
For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)
A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019
Also ordinal transform of A065331 (after the initial 0). - Antti Karttunen, Sep 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..19683

One less than A126759.
Cf. A003586, A003602, A007310, A064989, A249746, A253887, A254048, A273669, A322317, A323881 (Dirichlet inverse), A323882.
Cf. A347233 (Möbius transform) and also A349390, A349393, A349395 for other Dirichlet convolutions.
Ordinal transform of A065331 and of A322026 (after the initial 0).
Related arrays: A135765, A254102.

f[n_] := Block[{a}, a[0] = 0; a[1] = a[2] = a[3] = 1; a[x_] := Which[EvenQ@ x, a[x/2], Mod[x, 3] == 0, a[x/3], Mod[x, 6] == 1, 2 (x - 1)/6 + 1, Mod[x, 6] == 5, 2 (x - 5)/6 + 2]; Table[a@ i, {i, 0, n}]] (* Michael De Vlieger, Feb 03 2015 *)

A126760(n)={n&&n\=3^valuation(n,3)<M. F. Hasler, Jan 19 2016

a(n) = A126759(n)-1. [The original definition.]
From Antti Karttunen, Jan 28 2015: (Start)
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
Or with the last clause represented in another way:
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n-1) = 2n.
Other identities. For all n >= 1:
a(n) = A253887(A003602(n)).
a(6n-3) = a(4n-2) = a(2n-1) = A253887(n).
(End)
a(n) = A249746(A003602(A064989(n))). - Antti Karttunen, Feb 04 2015
a(n) = A323882(4*n). - Antti Karttunen, Apr 18 2022

Name replaced with an independent recurrence and the old description moved to the Formula section - Antti Karttunen, Jan 28 2015

A349134 Dirichlet inverse of Kimberling's paraphrases, A003602.

Original entry on oeis.org

1, -1, -2, 0, -3, 2, -4, 0, -1, 3, -6, 0, -7, 4, 4, 0, -9, 1, -10, 0, 5, 6, -12, 0, -4, 7, -2, 0, -15, -4, -16, 0, 7, 9, 6, 0, -19, 10, 8, 0, -21, -5, -22, 0, 3, 12, -24, 0, -9, 4, 10, 0, -27, 2, 8, 0, 11, 15, -30, 0, -31, 16, 4, 0, 9, -7, -34, 0, 13, -6, -36, 0, -37, 19, 8, 0, 9, -8, -40, 0, -4, 21, -42, 0, 11, 22

Offset: 1

Antti Karttunen, Nov 13 2021

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Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003602, A349135, A349136.

Cf. also A323881, A349125.

k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*k[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
A349134(n) = v349134[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A003602(n/d) * a(d).

a(n) = A349135(n) - A003602(n).

A347233 Möbius transform of A126760.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 5, 0, 6, 0, 0, 0, 7, 0, 7, 0, 0, 0, 9, 0, 10, 0, 0, 0, 8, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 15, 0, 14, 0, 0, 0, 17, 0, 14, 0, 0, 0, 19, 0, 20, 0, 0, 0, 16, 0, 22, 0, 0, 0, 23, 0, 24, 0, 0, 0, 20, 0, 26, 0, 0, 0, 27, 0, 22, 0, 0, 0, 29, 0, 24, 0, 0, 0, 24, 0, 32

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008683, A126760.
Cf. A000004, A349339 (even and odd bisection).
Cf. also A323881, A346485, A347234, A349136, A349391, A349392, A349393, A349395.

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A347233(n) = sumdiv(n,d,moebius(n/d)*A126760(d));

a(n) = Sum_{d|n} A008683(n/d) * A126760(d).

A323882 Sum of A126760 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, 1, 1, 4, 0, 1, 0, 6, 4, 1, 0, 1, 0, 2, 6, 8, 0, 1, 4, 10, 1, 3, 0, 0, 0, 1, 8, 12, 12, 1, 0, 14, 10, 2, 0, 0, 0, 4, 2, 16, 0, 1, 9, 14, 12, 5, 0, 1, 16, 3, 14, 20, 0, 2, 0, 22, 3, 1, 20, 0, 0, 6, 16, 12, 0, 1, 0, 26, 14, 7, 24, 0, 0, 2, 1, 28, 0, 3, 24, 30, 20, 4, 0, 2, 30, 8, 22, 32, 28, 1, 0, 25, 4, 9, 0, 0, 0, 5, 12

Offset: 1

Antti Karttunen, Feb 08 2019

From Antti Karttunen, Aug 18 2021: (Start)
No negative terms in range 1 .. 2^20.
Apparently zeros occur only on (some of the) positions given by A030059, with exceptions for example on n = 70, 105, 110, 130, 154, etc, where a(n) > 0.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A030059, A126760, A323881, A323884, A323885, A323887.

up_to = 20000;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA126760(n)));
A323881(n) = v323881[n];
A323882(n) = (A126760(n)+A323881(n));

a(n) = A126760(n) + A323881(n).

For n > 1, a(n) = -Sum_{d|n, 1A126760(d) * A323881(n/d). - Antti Karttunen, Aug 18 2021

A323886 Dirichlet inverse of A004718, Per Nørgård's "infinity sequence".

Original entry on oeis.org

1, 1, -2, 0, 0, -2, -3, 0, 2, 0, -1, 0, 1, -3, -4, 0, 0, 2, -3, 0, 11, -1, -2, 0, -3, 1, 0, 0, 2, -4, -5, 0, 2, 0, -1, 0, 1, -3, -8, 0, -1, 11, -2, 0, 16, -2, -3, 0, 10, -3, -4, 0, -2, 0, -1, 0, 8, 2, 1, 0, 3, -5, -26, 0, 0, 2, -3, 0, 7, -1, -2, 0, -3, 1, 12, 0, 8, -8, -5, 0, -5, -1, -2, 0, 0, -2, -11, 0, -2, 16, -7, 0, 21, -3, -4, 0, -3, 10, 0, 0, 2, -4, -5, 0

Offset: 1

Antti Karttunen, Feb 08 2019

sign

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A004718, A323881, A323887.

b[0] = 0;
b[n_?EvenQ] := b[n] = -b[n/2];
b[n_] := b[n] = b[(n - 1)/2] + 1;
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 16 2020 *)

up_to = 65537;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, v[n>>1]+1, -v[n/2])); (v); }; \\ After code in A004718.
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA004718list(up_to));
A323886(n) = v323886[n];

A323883 Dirichlet inverse of A322026.

Original entry on oeis.org

1, -2, -3, 0, -1, 7, -1, 2, 2, 2, -1, 0, -1, 2, 3, -1, -1, -6, -1, 0, 3, 2, -1, -11, 0, 2, 3, 0, -1, -7, -1, -1, 3, 2, 1, -1, -1, 2, 3, -2, -1, -7, -1, 0, -2, 2, -1, 7, 0, 0, 3, 0, -1, -9, 1, -2, 3, 2, -1, 0, -1, 2, -2, 3, 1, -7, -1, 0, 3, -2, -1, 20, -1, 2, 0, 0, 1, -7, -1, 1, -6, 2, -1, 0, 1, 2, 3, -2, -1, 6, 1, 0, 3, 2, 1, 8

Offset: 1

Antti Karttunen, Feb 08 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A007814, A007949, A322026, A323881, A323884.

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; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA323883(n) = v323883[n];

cp's OEIS Frontend

A353461 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A323881 (the Dirichlet inverse of A126760).

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A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

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A349134 Dirichlet inverse of Kimberling's paraphrases, A003602.

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A347233 Möbius transform of A126760.

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A323882 Sum of A126760 and its Dirichlet inverse.

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A323886 Dirichlet inverse of A004718, Per Nørgård's "infinity sequence".

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A323883 Dirichlet inverse of A322026.

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