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 1-4 of 4 results.

A356574 a(n) = Sum_{d|n} tau(d^4), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 6, 6, 15, 6, 36, 6, 28, 15, 36, 6, 90, 6, 36, 36, 45, 6, 90, 6, 90, 36, 36, 6, 168, 15, 36, 28, 90, 6, 216, 6, 66, 36, 36, 36, 225, 6, 36, 36, 168, 6, 216, 6, 90, 90, 36, 6, 270, 15, 90, 36, 90, 6, 168, 36, 168, 36, 36, 6, 540, 6, 36, 90, 91, 36, 216, 6, 90, 36, 216, 6, 420, 6, 36, 90, 90
Offset: 1

Views

Author

Seiichi Manyama, Dec 13 2022

Keywords

Links

Crossrefs

Cf. A007425, A035116, A061391, A358380, A359037, A359038.
Cf. A000005, A321348.

Programs

  • Mathematica
    Array[DivisorSum[#, DivisorSigma[0, #^4] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 2*e^2 + 3*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d^4));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n*d^2));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^2));
  • PARI
    a(n) = numdiv(n)*numdiv(n^2);
  • PARI
    my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^4)*x^k/(1-x^k)))
  • Python
    from math import prod
from sympy import factorint
def A356574(n): return prod((e+1)*((e<<1)+1) for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

Formula

a(n) = Sum_{d|n} tau(n * d^2) = Sum_{d|n} tau(n^2).
a(n) = tau(n) * tau(n^2).
G.f.: Sum_{k>=1} tau(k^4) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 2*e^2 + 3*e + 1. - Amiram Eldar, Dec 14 2022

A359037 a(n) = Sum_{d|n} tau(d^6), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 8, 8, 21, 8, 64, 8, 40, 21, 64, 8, 168, 8, 64, 64, 65, 8, 168, 8, 168, 64, 64, 8, 320, 21, 64, 40, 168, 8, 512, 8, 96, 64, 64, 64, 441, 8, 64, 64, 320, 8, 512, 8, 168, 168, 64, 8, 520, 21, 168, 64, 168, 8, 320, 64, 320, 64, 64, 8, 1344, 8, 64, 168, 133, 64, 512, 8, 168, 64, 512, 8, 840, 8
Offset: 1

Views

Author

Seiichi Manyama, Dec 13 2022

Keywords

Links

Crossrefs

Cf. A007425, A035116, A061391, A356574, A358380, A359038.
Cf. A000005, A321348.

Programs

  • Mathematica
    Array[DivisorSum[#, DivisorSigma[0, #^6] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 3*e^2 + 4*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d^6));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n*d^4));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^2*d^2));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^3));
  • PARI
    a(n) = numdiv(n)*numdiv(n^3);
  • PARI
    my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^6)*x^k/(1-x^k)))

Formula

a(n) = Sum_{d|n} tau(n * d^4) = Sum_{d|n} tau(n^2 * d^2) = Sum_{d|n} tau(n^3).
a(n) = tau(n) * tau(n^3).
G.f.: Sum_{k>=1} tau(k^6) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 3*e^2 + 4*e + 1. - Amiram Eldar, Dec 14 2022

A359038 a(n) = Sum_{d|n} tau(d^7), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 9, 9, 24, 9, 81, 9, 46, 24, 81, 9, 216, 9, 81, 81, 75, 9, 216, 9, 216, 81, 81, 9, 414, 24, 81, 46, 216, 9, 729, 9, 111, 81, 81, 81, 576, 9, 81, 81, 414, 9, 729, 9, 216, 216, 81, 9, 675, 24, 216, 81, 216, 9, 414, 81, 414, 81, 81, 9, 1944, 9, 81, 216, 154, 81, 729, 9, 216, 81, 729, 9
Offset: 1

Views

Author

Seiichi Manyama, Dec 13 2022

Keywords

Crossrefs

Cf. A000005, A007425, A035116, A061391, A356574, A358380, A359037.
Cf. A321348.

Programs

  • Mathematica
    Array[DivisorSum[#, DivisorSigma[0, #^7] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 7*e^2/2 + 9*e/2 + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d^7));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n*d^5));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^2*d^3));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^3*d));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^4/d));
  • PARI
    my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^7)*x^k/(1-x^k)))
  • Python
    from math import prod
from sympy import factorint
def A359038(n): return prod((e+1)*(7*e+2)>>1 for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

Formula

a(n) = Sum_{d|n} tau(n * d^5) = Sum_{d|n} tau(n^2 * d^3) = Sum_{d|n} tau(n^3 * d) = Sum_{d|n} tau(n^4 / d).
G.f.: Sum_{k>=1} tau(k^7) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 7*e^2/2 + 9*e/2 + 1. - Amiram Eldar, Dec 14 2022

A358380 a(n) = Sum_{d|n} tau(d^5), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 7, 7, 18, 7, 49, 7, 34, 18, 49, 7, 126, 7, 49, 49, 55, 7, 126, 7, 126, 49, 49, 7, 238, 18, 49, 34, 126, 7, 343, 7, 81, 49, 49, 49, 324, 7, 49, 49, 238, 7, 343, 7, 126, 126, 49, 7, 385, 18, 126, 49, 126, 7, 238, 49, 238, 49, 49, 7, 882, 7, 49, 126, 112, 49, 343, 7, 126, 49, 343, 7, 612, 7, 49, 126, 126
Offset: 1

Views

Author

Seiichi Manyama, Dec 13 2022

Keywords

Crossrefs

Cf. A000005, A007425, A035116, A061391, A356574, A359037, A359038.
Cf. A321348.

Programs

  • Mathematica
    Array[DivisorSum[#, DivisorSigma[0, #^5] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 5*e^2/2 + 7*e/2 + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d^5));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n*d^3));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^2*d));
  • PARI
    a(n) = sumdiv(n, d, numdiv(n^3/d));
  • PARI
    my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^5)*x^k/(1-x^k)))

Formula

a(n) = Sum_{d|n} tau(n * d^3) = Sum_{d|n} tau(n^2 * d) = Sum_{d|n} tau(n^3 / d).
G.f.: Sum_{k>=1} tau(k^5) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 5*e^2/2 + 7*e/2 + 1. - Amiram Eldar, Dec 14 2022
Showing 1-4 of 4 results.

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