A347955 -id:A347955 - cp's OEIS frontend

A349371 Inverse Möbius transform of Kimberling's paraphrases (A003602).

1, 2, 3, 3, 4, 6, 5, 4, 8, 8, 7, 9, 8, 10, 14, 5, 10, 16, 11, 12, 18, 14, 13, 12, 17, 16, 22, 15, 16, 28, 17, 6, 26, 20, 26, 24, 20, 22, 30, 16, 22, 36, 23, 21, 42, 26, 25, 15, 30, 34, 38, 24, 28, 44, 38, 20, 42, 32, 31, 42, 32, 34, 55, 7, 44, 52, 35, 30, 50, 52, 37, 32, 38, 40, 65, 33, 50, 60, 41, 20, 63, 44, 43

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

Dirichlet convolution of sigma (A000203) with A349431, or equally, A264740 with A349447. - Antti Karttunen, Nov 21 2021

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

Cf. A000203, A264740.
Cf. also A347954, A347955, A347956, A349136, A349370, A349372, A349373, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.
Cf. also A349393.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349371(n) = sumdiv(n,d,A003602(d));

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

a(n) = Sum_{d|n} A000203(n/d)*A349431(d) = Sum_{d|n} A264740(n/d)*A349447(d). - Antti Karttunen, Nov 21 2021

A347954 Dirichlet convolution of A003602 with A342001.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 6, 4, 11, 1, 20, 1, 14, 13, 10, 1, 26, 1, 29, 16, 20, 1, 37, 5, 23, 12, 38, 1, 81, 1, 15, 22, 29, 19, 62, 1, 32, 25, 55, 1, 106, 1, 56, 48, 38, 1, 59, 6, 48, 31, 65, 1, 74, 25, 73, 34, 47, 1, 191, 1, 50, 61, 21, 28, 156, 1, 83, 40, 151, 1, 112, 1, 59, 60, 92, 28, 181, 1, 89, 34, 65, 1, 254, 34

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

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

Cf. A003415, A003557, A003602, A342001.

Cf. also A347234, A347955, A347956.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003602(n) = (1+(n>>valuation(n,2)))/2;
A342001(n) = (A003415(n) / A003557(n));
A347954(n) = sumdiv(n,d,A003602(d)*A342001(n/d));

a(n) = Sum_{d|n} A003602(d) * A342001(n/d).

A347956 Dirichlet convolution of A003602 with A069359.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 7, 5, 11, 1, 22, 1, 14, 13, 15, 1, 35, 1, 31, 16, 20, 1, 50, 8, 23, 20, 40, 1, 81, 1, 31, 22, 29, 19, 95, 1, 32, 25, 71, 1, 106, 1, 58, 62, 38, 1, 106, 11, 77, 31, 67, 1, 134, 25, 92, 34, 47, 1, 217, 1, 50, 78, 63, 28, 156, 1, 85, 40, 151, 1, 215, 1, 59, 95, 94, 28, 181, 1, 151, 74, 65, 1, 286, 34

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

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

Cf. A003602, A069359.

Cf. also A347954, A347955, A347957.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347956(n) = sumdiv(n,d,A003602(d)*A069359(n/d));

a(n) = Sum_{d|n} A003602(d) * A069359(n/d).

A349370 Dirichlet convolution of Kimberling's paraphrases (A003602) with itself.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 8, 4, 14, 12, 12, 12, 14, 16, 28, 5, 18, 28, 20, 18, 38, 24, 24, 16, 35, 28, 48, 24, 30, 56, 32, 6, 58, 36, 60, 42, 38, 40, 68, 24, 42, 76, 44, 36, 108, 48, 48, 20, 66, 70, 88, 42, 54, 96, 92, 32, 98, 60, 60, 84, 62, 64, 148, 7, 108, 116, 68, 54, 118, 120, 72, 56, 74, 76, 176, 60, 126, 136, 80, 30

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A347954, A347955, A347956, A349136, A349371, A349372, A349373, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349370(n) = sumdiv(n,d,A003602(n/d)*A003602(d));

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

A349372 Dirichlet convolution of Kimberling's paraphrases (A003602) with tau (number of divisors, A000005).

Original entry on oeis.org

1, 3, 4, 6, 5, 12, 6, 10, 12, 15, 8, 24, 9, 18, 22, 15, 11, 36, 12, 30, 27, 24, 14, 40, 22, 27, 34, 36, 17, 66, 18, 21, 37, 33, 36, 72, 21, 36, 42, 50, 23, 81, 24, 48, 72, 42, 26, 60, 36, 66, 52, 54, 29, 102, 50, 60, 57, 51, 32, 132, 33, 54, 90, 28, 57, 111, 36, 66, 67, 108, 38, 120, 39, 63, 104, 72, 63, 126, 42, 75

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A000005, A003602.
Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349373, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.
Cf. also A349392.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349372(n) = sumdiv(n,d,A003602(n/d)*numdiv(d));

a(n) = Sum_{d|n} A003602(n/d) * A000005(d).

A349375 Dirichlet convolution of Kimberling's paraphrases (A003602) with Liouville's lambda.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 3, 0, 4, 0, 5, 1, 6, 0, 4, 1, 8, 0, 9, 2, 6, 0, 11, 0, 11, 0, 10, 3, 14, 0, 15, 0, 10, 0, 12, 4, 18, 0, 12, 0, 20, 0, 21, 5, 14, 0, 23, 1, 22, 0, 16, 6, 26, 0, 20, 0, 18, 0, 29, 4, 30, 0, 21, 1, 24, 0, 33, 8, 22, 0, 35, 0, 36, 0, 21, 9, 30, 0, 39, 2, 31, 0, 41, 6, 32, 0, 28, 0, 44, 0, 36, 11, 30, 0, 36

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A003602, A008836.
Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349372, A349373, A349374, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.
Cf. also A349395.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * LiouvilleLambda[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A008836(n) = ((-1)^bigomega(n));
A349375(n) = sumdiv(n,d,A003602(n/d)*A008836(d));

a(n) = Sum_{d|n} A003602(n/d) * A008836(d).

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

Original entry on oeis.org

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).

A347957 Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 3, 6, 1, 9, 1, 7, 7, 4, 1, 14, 1, 11, 8, 9, 1, 13, 4, 10, 8, 13, 1, 28, 1, 5, 10, 12, 9, 25, 1, 13, 11, 16, 1, 34, 1, 17, 22, 15, 1, 17, 5, 25, 13, 19, 1, 38, 11, 19, 14, 18, 1, 49, 1, 19, 26, 6, 12, 46, 1, 23, 16, 44, 1, 36, 1, 22, 31, 25, 12, 52, 1, 21, 22, 24, 1, 60, 14, 25, 19, 25, 1, 86

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Jon Maiga, Computer-generated formulas for A347957, Sequence Machine.

Cf. A001221, A003602, A023900, A181988, A205745.

Cf. also A328203, A347954, A347955, A347956.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A347957(n) = sumdiv(n,d,omega(n/d)*A003602(d));

a(n) = Sum_{d|n} A001221(n/d) * A003602(d).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. The first one is obvious, and even the second one should not be too hard to prove:
a(n) = Sum_{d|n} A023900(n/d) * A347956(d).
a(n) = Sum_{d|n} A181988(n/d) * A205745(d).
(End)

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

A349371 Inverse Möbius transform of Kimberling's paraphrases (A003602).

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A347954 Dirichlet convolution of A003602 with A342001.

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A347956 Dirichlet convolution of A003602 with A069359.

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A349370 Dirichlet convolution of Kimberling's paraphrases (A003602) with itself.

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A349372 Dirichlet convolution of Kimberling's paraphrases (A003602) with tau (number of divisors, A000005).

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A349375 Dirichlet convolution of Kimberling's paraphrases (A003602) with Liouville's lambda.

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

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A347957 Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).

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