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 1-8 of 8 results.

A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 7, 8, 9, 12, 11, 12, 13, 14, 18, 16, 17, 18, 19, 24, 21, 22, 23, 24, 31, 26, 27, 28, 29, 36, 31, 32, 33, 34, 42, 36, 37, 38, 39, 48, 41, 42, 43, 44, 54, 46, 47, 48, 49, 62, 51, 52, 53, 54, 66, 56, 57, 58, 59, 72, 61, 62, 63, 64, 78, 66, 67, 68, 69, 84, 71, 72, 73, 74, 93, 76
Offset: 1

Views

Author

Seiichi Manyama, Dec 16 2022

Keywords

Links

Crossrefs

Cf. A129527, A327625, A359099, A373188.
Cf. A000010, A055457.

Programs

  • Mathematica
    Array[DivisorSum[#, EulerPhi[5 #] &]/4 &, 76] (* Michael De Vlieger, Dec 16 2022 *)
f[p_, e_] := If[p == 5, (5^(e + 1) - 1)/4, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(5*d))/4;
  • PARI
    my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(5*k)*x^k/(1-x^k))/4)

Formula

G.f.: Sum_{k>=1} phi(5 * k) * x^k / (4 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(5^k) / (1 - x^(5^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = p if p != 5.
Dirichlet g.f.: zeta(s-1)*(1+1/(5^s-1)).
Sum_{k=1..n} a(k) ~ (25/48) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^5).
If n == 0 (mod 5), a(n) = n + a(n/5) otherwise a(n) = n. (End)

A359099 a(n) = (1/6) * Sum_{d|n} phi(7 * d).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 8, 9, 10, 11, 12, 13, 16, 15, 16, 17, 18, 19, 20, 24, 22, 23, 24, 25, 26, 27, 32, 29, 30, 31, 32, 33, 34, 40, 36, 37, 38, 39, 40, 41, 48, 43, 44, 45, 46, 47, 48, 57, 50, 51, 52, 53, 54, 55, 64, 57, 58, 59, 60, 61, 62, 72, 64, 65, 66, 67, 68, 69, 80, 71, 72, 73, 74, 75, 76, 88, 78, 79
Offset: 1

Views

Author

Seiichi Manyama, Dec 16 2022

Keywords

Links

Crossrefs

Cf. A129527, A327625, A359100, A373188.
Cf. A000010, A373217.

Programs

  • Mathematica
    Array[DivisorSum[#, EulerPhi[7 #] &]/6 &, 79] (* Michael De Vlieger, Dec 16 2022 *)
f[p_, e_] := If[p == 7, (7^(e + 1) - 1)/6, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(7*d))/6;
  • PARI
    my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(7*k)*x^k/(1-x^k))/6)

Formula

G.f.: Sum_{k>=1} phi(7 * k) * x^k / (6 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(7^k) / (1 - x^(7^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = p if p != 7.
Dirichlet g.f.: zeta(s-1)*(1+1/(7^s-1)).
Sum_{k=1..n} a(k) ~ (49/96) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^7).
If n == 0 (mod 7), a(n) = n + a(n/7) otherwise a(n) = n. (End)

A373188 Expansion of Sum_{k>=0} x^(4^k) / (1 - x^(4^k))^2.

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 10, 9, 10, 11, 15, 13, 14, 15, 21, 17, 18, 19, 25, 21, 22, 23, 30, 25, 26, 27, 35, 29, 30, 31, 42, 33, 34, 35, 45, 37, 38, 39, 50, 41, 42, 43, 55, 45, 46, 47, 63, 49, 50, 51, 65, 53, 54, 55, 70, 57, 58, 59, 75, 61, 62, 63, 85, 65, 66, 67, 85, 69, 70, 71, 90, 73, 74, 75, 95, 77, 78, 79, 105, 81, 82, 83, 105, 85, 86
Offset: 1

Views

Author

Seiichi Manyama, May 27 2024

Keywords

Crossrefs

Cf. A373187, A373189.
Cf. A129527, A327625, A359100.

Formula

G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^4).
a(4*n+1) = 4*n+1, a(4*n+2) = 4*n+2, a(4*n+3) = 4*n+3 and a(4*n+4) = 4*n+4 + a(n+1) for n >= 0.

A364222 Expansion of Sum_{k>=0} 3^k * x^(3^k) / (1 - x^(3^k))^2.

Original entry on oeis.org

1, 2, 6, 4, 5, 12, 7, 8, 27, 10, 11, 24, 13, 14, 30, 16, 17, 54, 19, 20, 42, 22, 23, 48, 25, 26, 108, 28, 29, 60, 31, 32, 66, 34, 35, 108, 37, 38, 78, 40, 41, 84, 43, 44, 135, 46, 47, 96, 49, 50, 102, 52, 53, 216, 55, 56, 114, 58, 59, 120, 61, 62, 189, 64, 65, 132, 67, 68, 138, 70, 71, 216, 73, 74
Offset: 1

Views

Author

Seiichi Manyama, Jul 14 2023

Keywords

Crossrefs

Cf. A007949, A051064, A327625.
Cf. A091512, A364223, A364224.

Programs

  • Mathematica
    a[n_] := n * (IntegerExponent[n, 3] + 1); Array[a, 100] (* Amiram Eldar, Jul 14 2023 *)
  • PARI
    a(n) = n*(valuation(n, 3)+1);

Formula

a(n) = n * A051064(n).
If n == 0 (mod 3), a(n) = n + 3 * a(n/3) otherwise a(n) = n.
From Amiram Eldar, Jul 14 2023: (Start)
Multiplicative with a(3^e) = (e+1)*3^e and a(p^e) = p*e if p != 3.
Dirichlet g.f.: (3^s/(3^s-3)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (3/4)*n^2. (End)

A373184 G.f. A(x) satisfies A(x) = 1/(1 - x)^2 - 1 + A(x^3).

Original entry on oeis.org

2, 3, 6, 5, 6, 10, 8, 9, 16, 11, 12, 18, 14, 15, 22, 17, 18, 29, 20, 21, 30, 23, 24, 34, 26, 27, 44, 29, 30, 42, 32, 33, 46, 35, 36, 55, 38, 39, 54, 41, 42, 58, 44, 45, 68, 47, 48, 66, 50, 51, 70, 53, 54, 84, 56, 57, 78, 59, 60, 82, 62, 63, 94, 65, 66, 90, 68, 69, 94, 71, 72, 107, 74, 75, 102, 77, 78, 106, 80, 81
Offset: 1

Views

Author

Seiichi Manyama, May 27 2024

Keywords

Links

Crossrefs

Cf. A084432, A373185.
Cf. A327625.

Programs

  • Ruby
    def A(k, n)
  ary = [0]
  (1..n).each{|i|
    j = i + 1
    j += ary[i / k] if i % k == 0
    ary << j
  }
  ary[1..-1]
end
p A(3, 80)

Formula

a(3*n+1) = 3*n+2, a(3*n+2) = 3*n+3 and a(3*n+3) = 3*n+4 + a(n+1) for n >= 0.
G.f.: A(x) = Sum_{k>=0} (1/(1 - x^(3^k))^2 - 1).

A364210 a(n) = (1/(2*n)) * Sum_{d|n} 3^(n/d-1) * phi(3*d).

Original entry on oeis.org

1, 2, 4, 8, 17, 44, 105, 278, 733, 1978, 5369, 14792, 40881, 113934, 318884, 896948, 2532161, 7174862, 20390553, 58114072, 166037460, 475473286, 1364393897, 3922640132, 11297181473, 32588043882, 94143179560, 272342824320, 788854912241, 2287679406940, 6641649422409
Offset: 1

Views

Author

Seiichi Manyama, Jul 13 2023

Keywords

Crossrefs

Cf. A000010, A034754, A195459, A327625.
Cf. A000013, A364211, A364212.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 3^(n/#-1)*EulerPhi[3*#]/(2*n) &]; Array[a, 30] (* Amiram Eldar, Jul 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 3^(n/d-1)*eulerphi(3*d))/(2*n);

Formula

G.f.: (-1/2) * Sum_{k>0} phi(3*k) * log(1-3*x^k)/(3*k).

A373186 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^3.

Original entry on oeis.org

1, 3, 7, 10, 15, 24, 28, 36, 52, 55, 66, 88, 91, 105, 135, 136, 153, 195, 190, 210, 259, 253, 276, 336, 325, 351, 430, 406, 435, 520, 496, 528, 627, 595, 630, 754, 703, 741, 871, 820, 861, 1008, 946, 990, 1170, 1081, 1128, 1312, 1225, 1275, 1479, 1378, 1431, 1680, 1540
Offset: 1

Views

Author

Seiichi Manyama, May 27 2024

Keywords

Crossrefs

Cf. A129527, A373187.
Cf. A327625, A338045.
Cf. A000217.

Formula

G.f. A(x) satisfies A(x) = x/(1 - x)^3 + A(x^3).
a(3*n+1) = A000217(3*n+1), a(3*n+2) = A000217(3*n+2) and a(3*n+3) = A000217(3*n+3) + a(n+1) for n >= 0.

A373397 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 11, 14, 13, 14, 15, 16, 17, 21, 19, 20, 21, 22, 23, 28, 25, 26, 27, 28, 29, 35, 31, 32, 33, 34, 35, 43, 37, 38, 39, 40, 41, 49, 43, 44, 45, 46, 47, 56, 49, 50, 51, 52, 53, 63, 55, 56, 57, 58, 59, 70, 61, 62, 63, 64, 65, 77, 67, 68, 69, 70, 71, 86, 73, 74, 75, 76, 77, 91
Offset: 1

Views

Author

Seiichi Manyama, Jun 04 2024

Keywords

Links

Crossrefs

Cf. A129527, A327625, A359099, A359100, A373188.
Cf. A373216.

Formula

G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^6).
If n == 0 (mod 6), a(n) = n + a(n/6) otherwise a(n) = n.
Showing 1-8 of 8 results.

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