A349433 -id:A349433 - cp's OEIS frontend

A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

1, -1, -1, -1, -2, 1, -3, -1, -1, 2, -5, 1, -6, 3, 4, -1, -8, 1, -9, 2, 6, 5, -11, 1, -2, 6, -1, 3, -14, -4, -15, -1, 10, 8, 12, 1, -18, 9, 12, 2, -20, -6, -21, 5, 4, 11, -23, 1, -3, 2, 16, 6, -26, 1, 20, 3, 18, 14, -29, -4, -30, 15, 6, -1, 24, -10, -33, 8, 22, -12, -35, 1, -36, 18, 4, 9, 30, -12, -39, 2, -1, 20

Offset: 1

Antti Karttunen, Nov 17 2021

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Dirichlet convolution of this sequence with A000010 gives A349136, which also proves the formula involving A023900.

Convolution with A000203 gives A349371.

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

Sequence A297381 negated.
Cf. A003602, A023900, A055615, A297381, A349432 (Dirichlet inverse), A349433 (sum with it).
Cf. also A000010, A000203, A349136, A349371, and also A349444, A349447.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, # * MoebiusMu [#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A055615(n) = (n*moebius(n));
A349431(n) = sumdiv(n,d,A003602(n/d)*A055615(d));

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A349431(n) = if(!bitand(n,n-1),A023900(n),A023900(n)/2);

a(n) = Sum_{d|n} A003602(n/d) * A055615(d).
a(n) = A023900(n) when n is a power of 2, and a(n) = A023900(n)/2 for all other numbers.
a(n) = -A297381(n).

A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 2, 2, 5, 2, 6, 3, 0, 8, 8, 2, 9, 4, 0, 5, 11, 4, 6, 6, 4, 6, 14, 0, 15, 16, 0, 8, 0, 4, 18, 9, 0, 8, 20, 0, 21, 10, -2, 11, 23, 8, 12, 6, 0, 12, 26, 4, 0, 12, 0, 14, 29, 0, 30, 15, -3, 32, 0, 0, 33, 16, 0, 0, 35, 8, 36, 18, -4, 18, 0, 0, 39, 16, 8, 20, 41, 0, 0, 21, 0, 20, 44, -2, 0, 22, 0, 23

Offset: 1

Antti Karttunen, Nov 17 2021

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

Cf. A003602, A055615, A349134, A349431 (Dirichlet inverse), A349433 (sum with it).

Cf. also A349445, A349448.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#] * k[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 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];
A003602(n) = (1+(n>>valuation(n,2)))/2;
A055615(n) = (n*moebius(n));
A349432(n) = sumdiv(n,d,d*A349134(n/d));

A349446 a(n) = A349444(n) + A349445(n).

Original entry on oeis.org

2, 0, 0, 1, 0, -2, 0, 1, 1, -4, 0, -1, 0, -6, 4, 1, 0, -5, 0, -2, 6, -10, 0, -1, 4, -12, 5, -3, 0, -4, 0, 1, 10, -16, 12, -2, 0, -18, 12, -2, 0, -6, 0, -5, 14, -22, 0, -1, 9, -16, 16, -6, 0, -13, 20, -3, 18, -28, 0, 0, 0, -30, 21, 1, 24, -10, 0, -8, 22, -12, 0, -2, 0, -36, 24, -9, 30, -12, 0, -2, 19, -40, 0, 0, 32

Offset: 1

Antti Karttunen, Nov 18 2021

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

Cf. A349444, A349445.

Cf. also A349433.

s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#]*k[n/#] &, # < n &]; a[n_] := DivisorSum[n,  s[#]*k[n/#] + IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A349446(n) = (A349444(n)+A349445(n)); \\ Needs also code from A349444 and A349445.

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349444(d) * A349445(n/d). [As the sequences are Dirichlet inverses of each other]

cp's OEIS Frontend

A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

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A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

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A349446 a(n) = A349444(n) + A349445(n).

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