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.

A350156 Inverse Moebius transform of A000056.

Original entry on oeis.org

1, 7, 25, 55, 121, 175, 337, 439, 673, 847, 1321, 1375, 2185, 2359, 3025, 3511, 4897, 4711, 6841, 6655, 8425, 9247, 12145, 10975, 15121, 15295, 18169, 18535, 24361, 21175, 29761, 28087, 33025, 34279, 40777, 37015, 50617, 47887, 54625, 53119, 68881, 58975, 79465, 72655, 81433, 85015
Offset: 1

Views

Author

Werner Schulte, Jan 19 2022

Keywords

Comments

Let f be an arbitrary arithmetic function. Define the sequence a(f; n) by a(f; n) = Sum_{i=1..n, k=1..n} f(n / gcd(gcd(i,k),n)) for n > 0. Then a(f; n) equals inverse Moebius transform of f(n) * A007434(n) for n > 0; if f is multiplicative then a(f; n) is multiplicative; this sequence uses f(n) = n (see formula section).

Crossrefs

Column k=2 of A372968.
Cf. A000010, A000056, A000578, A001158, A007434, A055615, A057660.
Cf. A372962, A372964.

Programs

  • Mathematica
    f[p_, e_] := p^(3*e) - (p - 1)*(p^(3*e) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Jan 19 2022 *)
  • Python
    from math import prod
from sympy import factorint
def A350156(n): return prod((q:=p**(3*e))-(p-1)*(q-1)//(p**3-1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Formula

Multiplicative with a(p^e) = p^(3*e) - (p-1) * (p^(3*e) - 1) / (p^3 - 1) for prime p and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-3) * zeta(s) / zeta(s-1).
a(n) = Sum_{i=1..n, k=1..n} n / gcd(gcd(i,k),n) for n > 0.
Dirichlet convolution with A000010 equals A000578.
Dirichlet convolution of A001158 and A055615.
Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/(360*zeta(3)) = 0.225098... . - Amiram Eldar, Oct 16 2022
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_2(d^2)/sigma(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2. - Seiichi Manyama, May 25 2024

A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).

Original entry on oeis.org

1, 15, 79, 239, 621, 1185, 2395, 3823, 6397, 9315, 14631, 18881, 28549, 35925, 49059, 61167, 83505, 95955, 130303, 148419, 189205, 219465, 279819, 302017, 388121, 428235, 518155, 572405, 707253, 735885, 923491, 978671, 1155849, 1252575, 1487295, 1528883
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Column k=3 of A372968.
Cf. A001159, A008683.
Cf. A013662, A013663.
Cf. A371492, A372962.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d,4));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_4(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1).
Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(5)/zeta(4) = 0.958057374... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_4(d^2)/sigma_2(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 2024

A371878 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} n/gcd(x_1, x_2, x_3, x_4, x_5, n).

Original entry on oeis.org

1, 63, 727, 4031, 15621, 45801, 117643, 257983, 529981, 984123, 1771551, 2930537, 4826797, 7411509, 11356467, 16510911, 24137553, 33388803, 47045863, 62968251, 85526461, 111607713, 148035867, 187553641, 244078121, 304088211, 386356147, 474218933, 594823293, 715457421
Offset: 1

Views

Author

Seiichi Manyama, May 25 2024

Keywords

Links

Crossrefs

Column k=5 of A372968.
Cf. A371491, A373007, A373103, A373105.
Cf. A013664, A013665.

Programs

  • Mathematica
    f[p_, e_] := (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, May 25 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 6));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, x_3, x_4, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_6(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(6) = 0.9911595361106... . (End)

A372961 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} n/gcd(x_1, x_2, x_3, x_4, n).

Original entry on oeis.org

1, 31, 241, 991, 3121, 7471, 16801, 31711, 58561, 96751, 161041, 238831, 371281, 520831, 752161, 1014751, 1419841, 1815391, 2476081, 3092911, 4049041, 4992271, 6436321, 7642351, 9753121, 11509711, 14230321, 16649791, 20511121, 23316991, 28629121, 32472031, 38810881
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Column k=4 of A372968.
Cf. A001160, A008683.
Cf. A013663, A013664.

Programs

  • Mathematica
    f[p_, e_] := (p^(5*e+5) - p^(5*e+1) + p - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 5));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+1) + p - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(5) = 0.981112769... . (End)
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, x_2, x_3, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A372969 a(n) = Sum_{1 <= x_1, x_2, ... , x_n <= n} n/gcd(x_1, x_2, ... , x_n, n).

Original entry on oeis.org

1, 7, 79, 991, 15621, 277495, 5764795, 133955071, 3486666301, 99951163687, 3138428376711, 106980008889391, 3937376385699277, 155563347996105679, 6568408050364922499, 295145653362359140351, 14063084452067724990993, 708233993284902846818911
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Main diagonal of A372968.
Cf. A108223.

Programs

  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, n+1));

Formula

a(n) = Sum_{d|n} mu(n/d) * n/d * sigma_{n+1}(d).
a(n) = Sum_{1 <= x_1, x_2, ... , x_n <= n} ( gcd(x_1, x_2, ... , x_{n-1}, n)/gcd(x_1, x_2, ... , x_n, n) )^n.
Showing 1-5 of 5 results.

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