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.

Previous Showing 11-20 of 20 results.

A343325 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k^5).

Original entry on oeis.org

1, 32, 243, 1520, 3125, 15552, 16807, 70496, 88452, 200000, 161051, 867024, 371293, 1075648, 1518750, 3164792, 1419857, 6609600, 2476099, 11150000, 8168202, 10307264, 6436343, 44875296, 14646875, 23762752, 31059855, 59967376, 20511149, 121500000, 28629151, 138957472, 78270786, 90870848
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 11 2021

Keywords

Crossrefs

Cf. A000584, A045778, A050368, A248884, A343285, A343322, A343323, A343324.

A328731 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^k.

Original entry on oeis.org

1, -2, -3, -1, -5, 0, -7, -4, -3, 0, -11, 3, -13, 0, 0, 3, -17, 6, -19, 5, 0, 0, -23, 18, -10, 0, -10, 7, -29, 30, -31, -2, 0, 0, 0, 24, -37, 0, 0, 30, -41, 42, -43, 11, 15, 0, -47, 27, -21, 20, 0, 13, -53, 38, 0, 42, 0, 0, -59, 60, -61, 0, 21, 17, 0, 66, -67, 17, 0, 70
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Comments

Dirichlet inverse of A050368.

Crossrefs

Cf. A006881 (positions of 0's), A050368, A316441, A328730.

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA050368(n/d) * a(d).

A344370 Dirichlet g.f.: Product_{k>=2} (1 + k^(1-s)).

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 7, 16, 9, 20, 11, 36, 13, 28, 30, 32, 17, 54, 19, 60, 42, 44, 23, 120, 25, 52, 54, 84, 29, 150, 31, 96, 66, 68, 70, 180, 37, 76, 78, 200, 41, 210, 43, 132, 135, 92, 47, 336, 49, 150, 102, 156, 53, 270, 110, 280, 114, 116, 59, 540
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2021

Keywords

Crossrefs

Cf. A022629, A045778, A050368, A050369, A224892, A344368, A344369.

Programs

  • Mathematica
    T[m_, 1] := 1; T[1, n_] := 0; T[1, 1] := 1; T[0, n_] := 0; T[m_, n_] := T[m, n] = Total[T[# - 1, n/#] & /@ Select[Divisors[n], # <= m &]]; A045778[n_] := T[n, n]; Table[n A045778[n], {n, 60}]

Formula

a(n) = n * A045778(n).

A328709 Dirichlet g.f.: Product_{k>=2} ((1 + k^(-s)) / (1 - k^(-s)))^k.

Original entry on oeis.org

1, 4, 6, 16, 10, 36, 14, 60, 36, 60, 22, 168, 26, 84, 90, 208, 34, 252, 38, 280, 126, 132, 46, 696, 100, 156, 200, 392, 58, 660, 62, 692, 198, 204, 210, 1296, 74, 228, 234, 1160, 82, 924, 86, 616, 630, 276, 94, 2640, 196, 700, 306, 728, 106, 1556, 330
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Comments

Dirichlet convolution of A050367 with A050368.

Crossrefs

Cf. A050367, A050368, A328707.

Formula

a(n) = Sum_{d|n} A050367(n/d) * A050368(d).

A328710 Dirichlet g.f.: (1/2) * Product_{i>=1, j>=1} (1 + (i*j)^(-s)).

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 10, 4, 8, 2, 22, 2, 8, 8, 19, 2, 22, 2, 22, 8, 8, 2, 56, 4, 8, 10, 22, 2, 40, 2, 38, 8, 8, 8, 71, 2, 8, 8, 56, 2, 40, 2, 22, 22, 8, 2, 128, 4, 22, 8, 22, 2, 56, 8, 56, 8, 8, 2, 132, 2, 8, 22, 73, 8, 40, 2, 22, 8, 40, 2, 202, 2, 8, 22, 22, 8, 40, 2, 128
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Crossrefs

Cf. A000005, A050368, A242649, A328711.

Formula

Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^tau(k), where tau = A000005.

A328711 Dirichlet g.f.: (1/2) * Product_{i>=1, j>=1} (1 + (i*j)^(-s))^i.

Original entry on oeis.org

1, 3, 4, 10, 6, 24, 8, 37, 19, 36, 12, 104, 14, 48, 48, 118, 18, 144, 20, 156, 64, 72, 24, 412, 46, 84, 96, 208, 30, 360, 32, 376, 96, 108, 96, 720, 38, 120, 112, 618, 42, 480, 44, 312, 288, 144, 48, 1500, 85, 339, 144, 364, 54, 792, 144, 824, 160, 180, 60, 1848
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Crossrefs

Cf. A000203, A050368, A242648, A328710.

Formula

Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^sigma(k), where sigma = A000203.

A328744 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 5, 8, 9, 13, 12, 23, 18, 27, 33, 39, 38, 63, 54, 80, 86, 101, 104, 161, 145, 183, 208, 254, 256, 361, 340, 435, 472, 550, 600, 776, 760, 918, 1018, 1221, 1260, 1576, 1610, 1929, 2129, 2408, 2590, 3172, 3274, 3833, 4173, 4783, 5120, 6054, 6414, 7414, 8025
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Comments

Number of ways to write n as an orderless product of orderless sums with distinct factors and each sum composed of distinct parts. Compare A318949.

Examples

			The a(4) = 2 ways: (4), (3+1).
The a(6) = 6 ways: (6), (4+2), (5+1), (3+2+1), (2)*(3), (2)*(2+1).

Links

Crossrefs

Cf. A000009, A045778, A050368, A318949.

Programs

  • PARI
    MultWeighT(u)={my(n=#u, v=vector(n, k, k==1)); for(k=2, n, if(u[k], my(m=logint(n,k), p=(1 + x + O(x*x^m))^u[k], w=vector(n)); for(i=0, m, w[k^i]=polcoef(p,i)); v=dirmul(v,w))); v}
seq(n)={MultWeighT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1))} \\ Andrew Howroyd, Oct 27 2019

A328876 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k + 1).

Original entry on oeis.org

1, 3, 4, 8, 6, 19, 8, 25, 16, 29, 12, 66, 14, 39, 40, 69, 18, 95, 20, 102, 54, 59, 24, 220, 41, 69, 72, 138, 30, 237, 32, 191, 82, 89, 84, 379, 38, 99, 96, 342, 42, 321, 44, 210, 206, 119, 48, 679, 78, 240, 124, 246, 54, 459, 128, 464, 138, 149, 60, 971
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 29 2019

Keywords

Comments

Number of ways to factor n into distinct factors with 3 kinds of 2, 4 kinds of 3, ..., k+1 kinds of k.
Dirichlet convolution of A045778 with A050368.

Crossrefs

Cf. A045778, A050368, A328851, A328877.

Programs

  • PARI
    seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, my(m=logint(n, k), p=(1 + x + O(x*x^m))^(k+1), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p, i)); v=dirmul(v, w)); v} \\ Andrew Howroyd, Oct 29 2019

Formula

a(n) = Sum_{d|n} A045778(n/d) * A050368(d).

A328877 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k - 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 6, 10, 9, 13, 10, 22, 12, 19, 22, 25, 16, 36, 18, 40, 32, 31, 22, 69, 30, 37, 42, 58, 28, 89, 30, 70, 52, 49, 58, 121, 36, 55, 62, 125, 40, 129, 42, 94, 108, 67, 46, 203, 63, 115, 82, 112, 52, 174, 94, 181, 92, 85, 58, 319, 60, 91, 156, 182
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 29 2019

Keywords

Comments

Number of ways to factor n into distinct factors with 1 kind of 2, 2 kinds of 3, ..., k-1 kinds of k.
Dirichlet convolution of A050368 with A316441.

Crossrefs

Cf. A050368, A316441, A328853, A328876.

Programs

  • PARI
    seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, my(m=logint(n, k), p=(1 + x + O(x*x^m))^(k-1), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p, i)); v=dirmul(v, w)); v} \\ Andrew Howroyd, Oct 29 2019

Formula

a(n) = Sum_{d|n} A050368(n/d) * A316441(d).

A344298 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(9^(k-1)).

Original entry on oeis.org

1, 9, 81, 765, 6561, 59778, 531441, 4789614, 43049961, 387479538, 3486784401, 31381653015, 282429536481, 2541870611298, 22876792986402, 205891175433096, 1853020188851841, 16677182091899187, 150094635296999121, 1350851721164795655, 12157665459099975522, 109418989162893418818
Offset: 1

Views

Author

Ilya Gutkovskiy, May 14 2021

Keywords

Crossrefs

Cf. A045778, A050368, A344253, A344265, A344266, A344267, A344268, A344269, A344270, A344271.

Programs

  • Mathematica
    dircon[v_, w_] := Module[{lv = Length[v], lw = Length[w], fv, fw}, fv[n_] := If[n <= lv, v[[n]], 0]; fw[n_] := If[n <= lw, w[[n]], 0]; Table[ DirichletConvolve[fv[n], fw[n], n, m], {m, Min[lv, lw]}]];
a[n_] := Module[{A, v, w, m}, If[n<1, 0, v = Table[Boole[k == 1], {k, n}]; For[k = 2, k <= n, k++, m = Length[IntegerDigits[n, k]] - 1; A = (1 + x)^(9^(k - 1)) + x*O[x]^m // Normal; w = Table[0, {n}]; For[i = 0, i <= m, i++, w[[k^i]] = Coefficient[A, x, i]]; v = dircon[v, w]]; v[[n]]]];
Array[a, 22] (* after Jean-François Alcover in A007896 *)
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.