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 1-6 of 6 results.

A347233 Möbius transform of A126760.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 5, 0, 6, 0, 0, 0, 7, 0, 7, 0, 0, 0, 9, 0, 10, 0, 0, 0, 8, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 15, 0, 14, 0, 0, 0, 17, 0, 14, 0, 0, 0, 19, 0, 20, 0, 0, 0, 16, 0, 22, 0, 0, 0, 23, 0, 24, 0, 0, 0, 20, 0, 26, 0, 0, 0, 27, 0, 22, 0, 0, 0, 29, 0, 24, 0, 0, 0, 24, 0, 32
Offset: 1

Views

Author

Antti Karttunen, Aug 26 2021

Keywords

Links

Crossrefs

Cf. A008683, A126760.
Cf. A000004, A349339 (even and odd bisection).
Cf. also A323881, A346485, A347234, A349136, A349391, A349392, A349393, A349395.

Programs

  • Mathematica
    f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A347233(n) = sumdiv(n,d,moebius(n/d)*A126760(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A126760(d).

A349390 Dirichlet convolution of A126760 with Kimberling's paraphrases, A003602.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 4, 8, 10, 10, 9, 12, 14, 17, 5, 15, 16, 17, 15, 24, 20, 20, 12, 28, 24, 22, 21, 25, 34, 27, 6, 35, 30, 47, 24, 32, 34, 42, 20, 35, 48, 37, 30, 50, 40, 40, 15, 54, 56, 53, 36, 45, 44, 71, 28, 60, 50, 50, 51, 52, 54, 71, 7, 84, 70, 57, 45, 71, 94, 60, 32, 62, 64, 100, 51, 99, 84, 67, 25, 63, 70, 70
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A003602, A126760.
Cf. A347233, A347234, A349391, A349392, A349393, A349395, A349431, A349444, A349447 for other Dirichlet convolutions of A126760. And also A349370.

Programs

  • Mathematica
    f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, f[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A003602(n) = (1+(n>>valuation(n,2)))/2;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349390(n) = sumdiv(n,d,A126760(n/d)*A003602(d));

Formula

a(n) = Sum_{d|n} A126760(n/d) * A003602(d).

A349393 Inverse Möbius transform of A126760.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 3, 6, 5, 6, 6, 8, 6, 5, 7, 6, 8, 9, 8, 10, 9, 8, 12, 12, 4, 12, 11, 12, 12, 6, 10, 14, 18, 9, 14, 16, 12, 12, 15, 16, 16, 15, 9, 18, 17, 10, 21, 24, 14, 18, 19, 8, 26, 16, 16, 22, 21, 18, 22, 24, 12, 7, 30, 20, 24, 21, 18, 36, 25, 12, 26, 28, 24, 24, 34, 24, 28, 15, 5, 30, 29, 24, 38, 32, 22
Offset: 1

Views

Author

Antti Karttunen, Nov 16 2021

Keywords

Links

Crossrefs

Cf. A347233, A347234, A349390, A349391, A349392, A349395 for other Dirichlet convolutions of A126760. And also A349371.

Programs

  • Mathematica
    f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349393(n) = sumdiv(n,d,A126760(d));
  • PARI
    a(n)=my(a=valuation(n,2),b=valuation(n,3),c=(a+1)*(b+1)); sumdiv(n/3^b>>a,d, d\6*2+d%3)*c; \\ Charles R Greathouse IV, Nov 16 2021

Formula

a(n) = Sum_{d|n} A126760(d).

A349395 Dirichlet convolution of A126760 with Liouville's lambda.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 4, 0, 0, 1, 5, 0, 6, 1, 0, 0, 7, 0, 8, 0, 0, 2, 9, 0, 10, 0, 0, 0, 8, 1, 12, 0, 0, 0, 13, 0, 14, 3, 1, 0, 15, 0, 15, 0, 0, 4, 17, 0, 14, 0, 0, 0, 19, 0, 20, 0, 2, 1, 16, 0, 22, 5, 0, 0, 23, 0, 24, 0, 0, 6, 20, 0, 26, 1, 1, 0, 27, 0, 22, 0, 0, 0, 29, 0, 24, 7, 0, 0, 24, 0, 32, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A008836, A126760.
Cf. A347233, A347234, A349390, A349391, A349392, A349393 for other Dirichlet convolutions of A126760. And also A349375.

Programs

  • Mathematica
    f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * LiouvilleLambda[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A008836(n) = ((-1)^bigomega(n));
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349395(n) = sumdiv(n,d,A126760(n/d)*A008836(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).

A349391 Dirichlet convolution of A126760 with omega.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 5, 1, 7, 1, 6, 5, 4, 1, 7, 1, 9, 6, 7, 1, 10, 3, 8, 3, 11, 1, 16, 1, 5, 7, 9, 7, 12, 1, 10, 8, 13, 1, 20, 1, 13, 9, 11, 1, 13, 4, 18, 9, 15, 1, 10, 8, 16, 10, 13, 1, 27, 1, 14, 11, 6, 9, 24, 1, 17, 11, 32, 1, 17, 1, 16, 18, 19, 9, 28, 1, 17, 4, 17, 1, 34, 10, 18, 13, 19, 1, 27, 10, 21
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A001221, A126760.
Cf. A347233, A347234, A349390, A349392, A349393, A349395 for other Dirichlet convolutions of A126760. And also A347957.

Programs

  • Mathematica
    f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * PrimeNu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349391(n) = sumdiv(n,d,A126760(n/d)*omega(d));

Formula

a(n) = Sum_{d|n} A126760(n/d) * A001221(d).
Showing 1-6 of 6 results.

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