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.

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A344205 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+4,5).

Original entry on oeis.org

1, 6, 21, 77, 126, 378, 462, 1184, 1518, 2758, 3003, 7497, 6188, 11340, 14274, 23154, 20349, 40734, 33649, 64218, 62832, 83798, 80730, 168756, 126756, 179634, 198709, 288358, 237336, 437694, 324632, 543964, 498960, 624036, 633969, 1088256, 749398, 1052562, 1092546
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2021

Keywords

Crossrefs

Cf. A000389, A000417, A001055, A050367, A329124, A344203, A344204, A344206, A344207.

A344206 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+5,6).

Original entry on oeis.org

1, 7, 28, 112, 210, 658, 924, 2388, 3409, 6475, 8008, 18746, 18564, 33600, 44640, 72408, 74613, 137746, 134596, 235655, 256102, 352066, 376740, 680260, 615930, 866229, 994336, 1400980, 1344904, 2172800, 1947792, 2929332, 2984905, 3784914, 4032420, 6128858, 5245786
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2021

Keywords

Crossrefs

Cf. A000428, A000579, A001055, A050367, A329124, A344203, A344204, A344205, A344207.

A344207 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+6,7).

Original entry on oeis.org

1, 8, 36, 156, 330, 1080, 1716, 4512, 7101, 14080, 19448, 43776, 50388, 91248, 128160, 209910, 245157, 431424, 480700, 800800, 949806, 1339624, 1560780, 2576376, 2684190, 3768960, 4512144, 6267472, 6724520, 10046160, 10295472, 14593272, 16081065, 20604816, 23048220
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2021

Keywords

Crossrefs

Cf. A000580, A001055, A050367, A255965, A329124, A344203, A344204, A344205, A344206.

A242649 Dirichlet g.f.: Product_{n>=2} 1/(1-1/n^s)^d(n), where d(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 14, 6, 8, 2, 26, 2, 8, 8, 33, 2, 26, 2, 26, 8, 8, 2, 72, 6, 8, 14, 26, 2, 40, 2, 70, 8, 8, 8, 95, 2, 8, 8, 72, 2, 40, 2, 26, 26, 8, 2, 184, 6, 26, 8, 26, 2, 72, 8, 72, 8, 8, 2, 148, 2, 8, 26, 149, 8, 40, 2, 26, 8, 40, 2, 282, 2, 8, 26
Offset: 1

Views

Author

N. J. A. Sloane, May 26 2014

Keywords

Links

Crossrefs

Cf. A000005, A007896, A050367, A242649.

Programs

  • PARI
    \\ Based on Michael Somos's code for A007896
n=101;
v = vector(n, k, k==1);
for(k=2, n, m = #digits(n, k) - 1; A = (1 - x)^ -(sigma(k,0)) + x * O(x^m); w = vector(n); for(i=0, m, w[k^i] = polcoeff(A, i)); v = dirmul(v, w));
v

A328745 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s))^p.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 4, 6, 10, 11, 9, 13, 14, 15, 5, 17, 12, 19, 15, 21, 22, 23, 12, 15, 26, 10, 21, 29, 30, 31, 6, 33, 34, 35, 18, 37, 38, 39, 20, 41, 42, 43, 33, 30, 46, 47, 15, 28, 30, 51, 39, 53, 20, 55, 28, 57, 58, 59, 45, 61, 62, 42, 7, 65, 66, 67, 51, 69, 70, 71, 24, 73, 74, 45
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Comments

Number of ways to factor n into 2 kinds of 2, 3 kinds of 3, 5 kinds of 5, ... , p kinds of p.

Links

Crossrefs

Cf. A008480, A050367, A059481, A135323, A182938.

Programs

  • Maple
    a:= n-> mul(binomial(i[1]+i[2]-1, i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 26 2019
  • Mathematica
    a[n_] := Times @@ (Binomial[#[[1]] + #[[2]] - 1, #[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^p)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

Formula

If n = Product (p_j^k_j) then a(n) = Product (binomial(p_j + k_j - 1, k_j)).
Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.40373... - Vaclav Kotesovec, Mar 28 2025

A328851 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k + 1).

Original entry on oeis.org

1, 3, 4, 11, 6, 19, 8, 34, 20, 29, 12, 78, 14, 39, 40, 104, 18, 107, 20, 120, 54, 59, 24, 277, 47, 69, 88, 162, 30, 237, 32, 299, 82, 89, 84, 478, 38, 99, 96, 429, 42, 321, 44, 246, 230, 119, 48, 921, 86, 258, 124, 288, 54, 535, 128, 581, 138, 149, 60, 1091
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2019

Keywords

Comments

Number of ways to factor n into 3 kinds of 2, 4 kinds of 3, ..., k+1 kinds of k.
Dirichlet convolution of A001055 with A050367.

Links

Crossrefs

Cf. A001055, A050367, A328853.

Programs

  • PARI
    seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, my(m=logint(n,k), p=1/(1 - x + O(x*x^m))^(1+k), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p,i)); v=dirmul(v,w)); v} \\ Andrew Howroyd, Oct 28 2019

Formula

a(n) = Sum_{d|n} A001055(n/d) * A050367(d).

A328853 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k - 1).

Original entry on oeis.org

1, 1, 2, 4, 4, 7, 6, 11, 11, 13, 10, 24, 12, 19, 22, 32, 16, 38, 18, 44, 32, 31, 22, 76, 34, 37, 46, 64, 28, 89, 30, 84, 52, 49, 58, 143, 36, 55, 62, 138, 40, 129, 42, 104, 116, 67, 46, 233, 69, 119, 82, 124, 52, 188, 94, 200, 92, 85, 58, 341, 60, 91, 168, 230
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2019

Keywords

Comments

Dirichlet convolution of A050367 with A114592.

Crossrefs

Cf. A001055, A050367, A114592, A328851.

Formula

a(n) = Sum_{d|n} A050367(n/d) * A114592(d).

A344368 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(1-s)).

Original entry on oeis.org

1, 2, 3, 8, 5, 12, 7, 24, 18, 20, 11, 48, 13, 28, 30, 80, 17, 72, 19, 80, 42, 44, 23, 168, 50, 52, 81, 112, 29, 150, 31, 224, 66, 68, 70, 324, 37, 76, 78, 280, 41, 210, 43, 176, 180, 92, 47, 576, 98, 200, 102, 208, 53, 378, 110, 392, 114, 116, 59, 660
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2021

Keywords

Crossrefs

Cf. A001055, A006906, A050367, A050369, A224892, A344369, A344370.

Programs

  • Mathematica
    T[, 1] = T[1, ] = 1; T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] T[n/#, #] &]; A001055[n_] := T[n, n]; Table[n A001055[n], {n, 60}]

Formula

a(n) = n * A001055(n).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A224892(n/d) * a(d).

A242648 Dirichlet g.f.: Product_{n>=2} 1/(1-1/n^s)^sigma(n).

Original entry on oeis.org

1, 3, 4, 13, 6, 24, 8, 46, 23, 36, 12, 116, 14, 48, 48, 161, 18, 156, 20, 174, 64, 72, 24, 484, 52, 84, 112, 232, 30, 360, 32, 526, 96, 108, 96, 841, 38, 120, 112, 726, 42, 480, 44, 348, 312, 144, 48, 1864, 93, 357, 144, 406, 54, 888, 144, 968, 160, 180
Offset: 1

Views

Author

N. J. A. Sloane, May 26 2014

Keywords

Crossrefs

Cf. A000203, A050367, A007896, A242649.

Programs

  • PARI
    \\ Based on Michael Somos's code for A007896
n=101;
v = vector(n, k, k==1);
   for(k=2, n, m = #digits(n, k) - 1; A = (1 - x)^ -(sigma(k)) + x * O(x^m); w = vector(n); for(i=0, m, w[k^i] = polcoeff(A, i)); v = dirmul(v, w));
v

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).
Previous Showing 21-30 of 32 results. Next

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

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