A349432 -id:A349432 - cp's OEIS frontend

A349433 a(n) = A349431(n) + A349432(n).

2, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 3, 0, 6, 4, 7, 0, 3, 0, 6, 6, 10, 0, 5, 4, 12, 3, 9, 0, -4, 0, 15, 10, 16, 12, 5, 0, 18, 12, 10, 0, -6, 0, 15, 2, 22, 0, 9, 9, 8, 16, 18, 0, 5, 20, 15, 18, 28, 0, -4, 0, 30, 3, 31, 24, -10, 0, 24, 22, -12, 0, 9, 0, 36, 0, 27, 30, -12, 0, 18, 7, 40, 0, -6, 32, 42, 28, 25, 0, -6, 36, 33

Offset: 1

Antti Karttunen, Nov 17 2021

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

Cf. A349431, A349432.

Cf. also A349446.

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/#] + # * MoebiusMu[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A349433(n) = (A349431(n) + A349432(n)); \\ Needs also code from A349431 and A349432.

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A349431(d) * A349432(n/d). [As the sequences are Dirichlet inverses of each other]

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

Original entry on oeis.org

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).

A349445 Dirichlet convolution of A001511 (the 2-adic valuation of 2n) with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2021

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

Cf. A001511, A003602, A349134, A349444 (Dirichlet inverse), A349446 (sum with it).

Cf. also A349432, 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, IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A001511(n) = (1+valuation(n,2));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349445(n) = sumdiv(n,d,A001511(n/d)*A349134(d));

a(n) = Sum_{d|n} A001511(n/d) * A349134(d).

If p odd prime, a(p) = (1-p)/2. - Bernard Schott, Nov 19 2021

A349448 Dirichlet convolution of A000265 (odd part of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 6, 0, 0, 0, 8, 0, 9, 0, 0, 0, 11, 0, 6, 0, 4, 0, 14, 0, 15, 0, 0, 0, 0, 0, 18, 0, 0, 0, 20, 0, 21, 0, -2, 0, 23, 0, 12, 0, 0, 0, 26, 0, 0, 0, 0, 0, 29, 0, 30, 0, -3, 0, 0, 0, 33, 0, 0, 0, 35, 0, 36, 0, -4, 0, 0, 0, 39, 0, 8, 0, 41, 0, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, 0, 0, 48, 0

Offset: 1

Antti Karttunen, Nov 19 2021

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

Cf. A000265, A003602, A349134, A349447 (Dirichlet inverse).

Cf. also A349432, A349445.

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, # / 2^IntegerExponent[#, 2] * kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349448(n) = sumdiv(n,d,A000265(d)*A349134(n/d));

a(n) = Sum_{d|n} A000265(d) * A349134(n/d).
From Bernard Schott, Dec 18 2021: (Start)
If p is an odd prime, a(p) = (p-1)/2.
If n is even, a(n) = 0. (End)

A349380 Dirichlet convolution of A003415 (arithmetic derivative of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 8, 4, 3, 1, 5, 1, 4, 3, 20, 1, 6, 1, 8, 4, 6, 1, 12, 7, 7, 14, 11, 1, 3, 1, 48, 6, 9, 5, 14, 1, 10, 7, 20, 1, 4, 1, 17, 8, 12, 1, 28, 10, 13, 9, 20, 1, 18, 7, 28, 10, 15, 1, 6, 1, 16, 11, 112, 8, 6, 1, 26, 12, 5, 1, 32, 1, 19, 11, 29, 8, 7, 1, 48, 46, 21, 1, 8, 10, 22, 15, 44, 1, 6, 9, 35, 16

Offset: 1

Antti Karttunen, Nov 21 2021

Dirichlet convolution of A349394 with A349432.

Dirichlet convolution with A349136 gives A300251.

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

Cf. A003415, A003602, A349134, A349394, A349432.

Cf. also A300251, A347955, A349136, A349431.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349380(n) = sumdiv(n,d,A003415(d)*A349134(n/d));

a(n) = Sum_{d|n} A003415(n/d) * A349134(d).

a(n) = Sum_{d|n} A349394(n/d) * A349432(d).

A353460 Dirichlet convolution of A126760 with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, 0, -2, 0, -2, 0, -2, 0, -1, 0, -3, 0, -3, 0, -2, 0, -4, 0, -1, 0, -4, 0, -5, 0, -5, 0, -3, 0, 1, 0, -6, 0, -4, 0, -7, 0, -7, 0, 0, 0, -8, 0, -4, 0, -5, 0, -9, 0, 3, 0, -6, 0, -10, 0, -10, 0, -1, 0, 2, 0, -11, 0, -7, 0, -12, 0, -12, 0, -3, 0, 1, 0, -13, 0, -8, 0, -14, 0, 4, 0, -9, 0, -15, 0, 0, 0

Offset: 1

Antti Karttunen, Apr 20 2022

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

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

Cf. A003602, A126760, A349134, A353461 (Dirichlet inverse), A353462 (sum with it).

Cf. also A349371, A349380, A349393, A349432.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A353460(n) = sumdiv(n,d,A126760(d)*A349134(n/d));

a(n) = Sum_{d|n} A126760(d) * A349134(n/d).

cp's OEIS Frontend

A349433 a(n) = A349431(n) + A349432(n).

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

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A349445 Dirichlet convolution of A001511 (the 2-adic valuation of 2n) with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

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A349448 Dirichlet convolution of A000265 (odd part of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

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A349380 Dirichlet convolution of A003415 (arithmetic derivative of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

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A353460 Dirichlet convolution of A126760 with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

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