A349446 -id:A349446 - cp's OEIS frontend

A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).

1, -1, 1, 0, 2, -1, 3, 0, 3, -2, 5, 0, 6, -3, 4, 0, 8, -3, 9, 0, 6, -5, 11, 0, 10, -6, 9, 0, 14, -4, 15, 0, 10, -8, 12, 0, 18, -9, 12, 0, 20, -6, 21, 0, 12, -11, 23, 0, 21, -10, 16, 0, 26, -9, 20, 0, 18, -14, 29, 0, 30, -15, 18, 0, 24, -10, 33, 0, 22, -12, 35, 0, 36, -18, 20, 0, 30, -12, 39, 0, 27, -20, 41, 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. A001511, A003602, A008683, A092673, A349445 (Dirichlet inverse), A349446 (sum with it).

Cf. also A349431, A349447.

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

A003602(n) = (1+(n>>valuation(n,2)))/2;
A092673(n) = if(n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2))); \\ From A092673
A349444(n) = sumdiv(n,d,A003602(n/d)*A092673(d));

a(n) = Sum_{d|n} A003602(n/d) * A092673(d).

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

sign

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

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

Original entry on oeis.org

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]

cp's OEIS Frontend

A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).

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

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