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.

A344221 a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 6, 13, 8, 30, 10, 29, 21, 40, 14, 78, 16, 50, 48, 61, 20, 105, 22, 104, 60, 70, 26, 174, 43, 80, 66, 130, 32, 240, 34, 125, 84, 100, 80, 273, 40, 110, 96, 232, 44, 300, 46, 182, 168, 130, 50, 366, 73, 215, 120, 208, 56, 330, 112, 290, 132, 160, 62, 624, 64, 170, 210, 253, 128, 420
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Links

Crossrefs

Cf. A060648, A072691, A344222, A344223, A344224, A344225.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[0,GCD[k,n]^3],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 2) - 3)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a(n) = sum(k=1, n, numdiv(gcd(k, n)^3));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^3));
  • PARI
    a(n) = n*sumdiv(n, d, 3^omega(d)/d);

Formula

a(n) = Sum_{d|n} phi(n/d) * tau(d^3).
a(n) = n * Sum_{d|n} 3^omega(d) / d.
If p is prime, a(p) = 3 + p.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p^e*(p + 2) - 3)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2) = 1.8019184198... . (End)

A344222 a(n) = Sum_{k=1..n} tau(gcd(k,n)^4), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 6, 7, 16, 9, 42, 11, 36, 25, 54, 15, 112, 17, 66, 63, 76, 21, 150, 23, 144, 77, 90, 27, 252, 49, 102, 79, 176, 33, 378, 35, 156, 105, 126, 99, 400, 41, 138, 119, 324, 45, 462, 47, 240, 225, 162, 51, 532, 81, 294, 147, 272, 57, 474, 135, 396, 161, 198, 63, 1008, 65, 210, 275, 316, 153, 630, 71
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Links

Crossrefs

Cf. A060648, A072691, A344221, A344223, A344224, A344225.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[0,GCD[k,n]^4],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 3) - 4)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a(n) = sum(k=1, n, numdiv(gcd(k, n)^4));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^4));
  • PARI
    a(n) = n*sumdiv(n, d, 4^omega(d)/d);

Formula

a(n) = Sum_{d|n} phi(n/d) * tau(d^4).
a(n) = n * Sum_{d|n} 4^omega(d) / d.
If p is prime, a(p) = 4 + p.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p^e*(p + 3) - 4)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 3/p^2) = 2.4997873122... . (End)

A344224 a(n) = Sum_{k=1..n} tau(gcd(k,n)^gcd(k,n)), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 4, 6, 14, 10, 61, 14, 44, 33, 143, 22, 410, 26, 257, 292, 128, 34, 852, 38, 1050, 536, 581, 46, 2352, 95, 791, 162, 1978, 58, 30545, 62, 352, 1240, 1307, 1388, 6918, 74, 1613, 1700, 6264, 82, 80823, 86, 4698, 4866, 2321, 94, 12416, 189, 5790, 2836, 6490, 106, 10881, 3284, 12032, 3512, 3623
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Links

Crossrefs

Cf. A344221, A344222, A344223, A344225.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[0,GCD[k,n]^(GCD[k,n])],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
  • PARI
    a(n) = sum(k=1, n, numdiv(gcd(k, n)^gcd(k,n)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^d));

Formula

a(n) = Sum_{d|n} phi(n/d) * tau(d^d).
If p is prime, a(p) = 2*p.

A344225 a(n) = Sum_{k=1..n} tau(gcd(k,n)^(n/gcd(k,n))), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 3, 4, 8, 6, 17, 8, 23, 17, 35, 12, 57, 14, 61, 44, 74, 18, 123, 20, 121, 72, 137, 24, 215, 47, 187, 96, 217, 30, 387, 32, 261, 152, 311, 100, 488, 38, 385, 204, 441, 42, 813, 44, 505, 294, 557, 48, 961, 93, 739, 332, 697, 54, 1333, 188, 787, 408, 875, 60, 1723, 62, 997, 488, 976, 244, 2433, 68
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Links

Crossrefs

Cf. A344221, A344222, A344223, A344224.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[0,GCD[k,n]^(n/GCD[k,n])],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
  • PARI
    a(n) = sum(k=1, n, numdiv(gcd(k, n)^(n/gcd(k, n))));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^(n/d)));

Formula

a(n) = Sum_{d|n} phi(n/d) * tau(d^(n/d)).
If p is prime, a(p) = 1 + p.

A344226 a(n) = Sum_{d|n} n^omega(d) / d.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 5, 18, 2, 50, 2, 24, 24, 16, 2, 90, 2, 80, 32, 36, 2, 198, 7, 42, 14, 110, 2, 1232, 2, 32, 48, 54, 48, 476, 2, 60, 56, 324, 2, 2310, 2, 170, 210, 72, 2, 782, 9, 338, 72, 200, 2, 756, 72, 450, 80, 90, 2, 12558, 2, 96, 290, 64, 84, 5474, 2, 260, 96, 5940
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Links

Crossrefs

Cf. A062319, A344223.

Programs

  • Mathematica
    Table[DivisorSum[n,n^PrimeNu@#/#&],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
  • PARI
    a(n) = sumdiv(n, d, n^omega(d)/d);
  • PARI
    a(n) = sum(k=1, n, numdiv(gcd(k, n)^n))/n;
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^n))/n;

Formula

a(n) = (1/n) * Sum_{k=1..n} tau(gcd(k,n)^n).
a(n) = (1/n) * Sum_{d|n} phi(n/d) * tau(d^n).
If p is prime, a(p) = 2.

A373002 a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^n ).

Original entry on oeis.org

1, 4, 9, 24, 25, 120, 49, 144, 135, 540, 121, 1728, 169, 1456, 2145, 800, 289, 5220, 361, 8840, 5985, 5544, 529, 21216, 1125, 9100, 1863, 25200, 841, 252000, 961, 4160, 23529, 20196, 31465, 94392, 1369, 28120, 38961, 113520, 1681, 991452, 1849, 101024, 118215
Offset: 1

Views

Author

Seiichi Manyama, May 19 2024

Keywords

Crossrefs

Cf. A062380, A062949, A372997, A372999.
Cf. A344223, A373003.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d^n));

Formula

a(n) = n * A373003(n).
If p is prime, a(p) = p^2.
a(n) = Sum_{d|n} phi(d) * tau(d^n).
Showing 1-6 of 6 results.

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