A349431 -id:A349431 - cp's OEIS frontend

A349374 Dirichlet convolution of Kimberling's paraphrases (A003602) with squarefree part of n (A007913).

1, 3, 5, 4, 8, 15, 11, 6, 12, 24, 17, 20, 20, 33, 42, 7, 26, 36, 29, 32, 58, 51, 35, 30, 29, 60, 34, 44, 44, 126, 47, 9, 90, 78, 94, 48, 56, 87, 106, 48, 62, 174, 65, 68, 110, 105, 71, 35, 54, 87, 138, 80, 80, 102, 146, 66, 154, 132, 89, 168, 92, 141, 153, 10, 172, 270, 101, 104, 186, 282, 107, 72, 110, 168, 167, 116

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A003602, A007913.

Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349372, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.

f[p_, e_] := p^Mod[e, 2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349374(n) = sumdiv(n,d,A003602(n/d)*core(d));

a(n) = Sum_{d|n} A003602(n/d) * A007913(d).

A349373 Dirichlet convolution of Kimberling's paraphrases (A003602) with Dirichlet inverse of Euler phi (A023900).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, -2, -2, -1, 0, -4, 0, -5, 0, 2, -3, -7, 0, -8, 1, 3, 0, -10, 0, -3, 0, -2, 2, -13, 0, -14, -4, 5, 0, 8, 1, -17, 0, 6, 2, -19, 0, -20, 4, 5, 0, -22, 0, -5, 0, 8, 5, -25, 0, 14, 4, 9, 0, -28, -2, -29, 0, 8, -5, 17, 0, -32, 7, 11, 0, -34, 2, -35, 0, 4, 8, 23, 0, -38, 3, -3, 0, -40, -3, 23, 0, 14, 8

Offset: 1

Antti Karttunen, Nov 15 2021

sign

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

Cf. A003602, A023900.

Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349372, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.

f[p_, e_] := (1 - p); d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A349373(n) = sumdiv(n,d,A003602(n/d)*A023900(d));

a(n) = Sum_{d|n} A003602(n/d) * A023900(d).

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).

cp's OEIS Frontend

A349374 Dirichlet convolution of Kimberling's paraphrases (A003602) with squarefree part of n (A007913).

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A349373 Dirichlet convolution of Kimberling's paraphrases (A003602) with Dirichlet inverse of Euler phi (A023900).

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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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