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 1-5 of 5 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

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).

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 1-5 of 5 results.

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