cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A003602(n) = (1+(n>>valuation(n,2)))/2;
A349371(n) = sumdiv(n,d,A003602(d));

Formula

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

Views

Author

Antti Karttunen, Sep 20 2021

Keywords

Links

Crossrefs

Cf. A003415, A003557, A003602, A342001.
Cf. also A347234, A347955, A347956.

Programs

Formula

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

A347955 Dirichlet convolution of A003415 (arithmetic derivative) with A003602 (Kimberling's paraphrases).

Original entry on oeis.org

0, 1, 1, 5, 1, 8, 1, 17, 8, 11, 1, 32, 1, 14, 13, 49, 1, 44, 1, 47, 16, 20, 1, 100, 13, 23, 44, 62, 1, 81, 1, 129, 22, 29, 19, 156, 1, 32, 25, 151, 1, 106, 1, 92, 86, 38, 1, 276, 18, 92, 31, 107, 1, 206, 25, 202, 34, 47, 1, 301, 1, 50, 111, 321, 28, 156, 1, 137, 40, 151, 1, 460, 1, 59, 120, 152, 28, 181, 1, 423, 206
Offset: 1

Views

Author

Antti Karttunen, Sep 20 2021

Keywords

Links

Crossrefs

Cf. A003415, A003602.
Cf. also A347954, A347956.

Programs

Formula

a(n) = Sum_{d|n} A003415(n/d) * A003602(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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A003602(n) = (1+(n>>valuation(n,2)))/2;
A349370(n) = sumdiv(n,d,A003602(n/d)*A003602(d));

Formula

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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * LiouvilleLambda[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A003602(n) = (1+(n>>valuation(n,2)))/2;
A008836(n) = ((-1)^bigomega(n));
A349375(n) = sumdiv(n,d,A003602(n/d)*A008836(d));

Formula

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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Antti Karttunen, Sep 20 2021

Keywords

Links

Crossrefs

Cf. A001221, A003602, A023900, A181988, A205745.
Cf. also A328203, A347954, A347955, A347956.

Programs

Formula

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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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));

Formula

a(n) = Sum_{d|n} A003602(n/d) * A023900(d).
Showing 1-9 of 9 results.

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