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 41-45 of 45 results.

A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

Original entry on oeis.org

1, 10, 20, 67, 52, 200, 100, 380, 282, 520, 244, 1340, 340, 1000, 1040, 1973, 580, 2820, 724, 3484, 2000, 2440, 1060, 7600, 1978, 3400, 3460, 6700, 1684, 10400, 1924, 9710, 4880, 5800, 5200, 18894, 2740, 7240, 6800, 19760, 3364, 20000, 3700, 16348, 14664
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

Comments

Dirichlet convolution of A001157 with itself.
Dirichlet convolution of A000005 with A034714.
Dirichlet convolution of A000290 with A007433.

Crossrefs

Cf. A000005, A000290, A001157, A001620, A007426, A007433, A034714, A034761.
Cf. A175705.

Programs

  • Magma
    [&+[DivisorSigma(2,d)*DivisorSigma(2, n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Oct 16 2019
  • Mathematica
    Table[DivisorSum[n, DivisorSigma[2, #] DivisorSigma[2, n/#] &], {n, 1, 45}]
f[p_, e_] :=((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3 ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 / (1 - p^2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

Formula

a(n) = Sum_{d|n} sigma_2(d) * sigma_2(n/d), where sigma_2 = A001157.
a(n) = Sum_{d|n} d^2 * tau(d) * tau(n/d), where tau = A000005.
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 * (zeta(3)*(log(n)/3 + 2*gamma/3 - 1/9) + 2*zeta'(3)/3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(p^e) = ((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3. - Amiram Eldar, Sep 15 2023

A341636 a(n) = Sum_{d|n} phi(d) * tau(d) * tau(n/d).

Original entry on oeis.org

1, 4, 6, 13, 10, 24, 14, 38, 29, 40, 22, 78, 26, 56, 60, 103, 34, 116, 38, 130, 84, 88, 46, 228, 79, 104, 124, 182, 58, 240, 62, 264, 132, 136, 140, 377, 74, 152, 156, 380, 82, 336, 86, 286, 290, 184, 94, 618, 153, 316, 204, 338, 106, 496, 220, 532, 228, 232, 118, 780, 122, 248
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

Comments

Inverse Moebius transform of A062949.

Links

Crossrefs

Cf. A000005, A000010, A000203, A007426, A055507, A062355, A062949.

Programs

  • Mathematica
    Table[Sum[EulerPhi[d] DivisorSigma[0, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 62}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[0, n/GCD[n, k]], {k, n}], {n, 62}]
f[p_, e_] := (p + 1 + e*(p - 1) + p^(e + 1)*(e*(p - 1) + p - 3))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

Formula

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * tau(n/gcd(n,k)).
a(n) = Sum_{d|n} A062949(d).
Multiplicative with a(p^e) = (p + 1 + e*(p-1) + p^(e+1)*(e*(p-1)+p-3))/(p-1)^2. - Amiram Eldar, Sep 15 2023

A098530 T(n,k) counts solid partitions of n+1 that can be 'shrunk' in k ways to a solid partition of n by removing 1 element from it. Equivalently, it counts how many solid partitions of n+1 have k different solid partitions of n it just covers.

Original entry on oeis.org

4, 4, 6, 10, 12, 4, 4, 42, 12, 1, 16, 60, 60, 4, 4, 105, 164, 34, 20, 162, 316, 180, 6, 10, 202, 672, 484, 96
Offset: 1

Views

Author

Wouter Meeussen, Sep 12 2004

Keywords

Comments

Sequence starts 4; 4,6; 10,12,4; 4,42,12,1; 16,60,60,4; 4,105,164,34; Row sums are A000293= the solid partitions of n+1 apart from offset. First column conjectured to be the (beheaded) A007426.

Examples

			T(3,3)=4 because the only solid partitions of 3+1=4 that can be shrunk in exactly 3 ways to plane partitions of 3 are
[{{2,1},{1}}], [{{2,1}},{{1}}], [{{2},{1}},{{1}}] and [{{1,1},{1}},{{1}}].

Crossrefs

Cf. A000293, A007426, A098529.

Programs

  • Mathematica
    (* functions 'solidform' and 'coverssolidQ', see A098052 *) Table[Frequencies[Count[Flatten[solidform / @ Partitions[n+1]], q_/;coverssolidQ[q, # ]]&/ @ Flatten[solidform / @ Partitions[n]]], {n, 1, 8}]

A140704 A051731^3 * A000012.

Original entry on oeis.org

1, 4, 1, 4, 1, 1, 10, 4, 1, 1, 4, 1, 1, 1, 1, 16, 7, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 20, 10, 4, 4, 1, 1, 1, 1, 10, 4, 4, 1, 1, 1, 1, 1, 1, 16, 7, 4, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 40, 22, 13, 7, 4, 4, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, May 24 2008

Keywords

Comments

Row sums = A007430: (1, 5, 6, 16, 8, 30, 10,...).
Left column = A007426: (1, 4, 4, 10, 4, 16, 4,...).

Examples

			First few rows of the triangle are:
1;
4, 1;
4, 1, 1;
10, 4, 1, 1;
4, 1, 1, 1, 1;
16, 7, 4, 1, 1, 1;
4, 1, 1, 1, 1, 1, 1;
20, 10, 4, 4, 1, 1, 1, 1;
10, 4, 4, 1, 1, 1, 1, 1, 1;
...

Crossrefs

Cf. A051731, A007430, A007426.

Formula

A051731^3 * A000012 as infinite lower triangular matrices, where A051731 = the inverse Mobius transform and A000012 an infinite lower triangular matrix with all 1's.

A327774 Composite numbers m such that tau_k(m) = m for some k, where tau_k is the k-th Piltz divisor function (A077592).

Original entry on oeis.org

18, 36, 75, 100, 200, 224, 225, 441, 560, 1183, 1344, 1920, 3025, 8281, 26011, 34606, 64009, 72030, 76895, 115351, 197173, 280041, 494209, 538265, 1168561, 1947271, 2927521, 3575881, 3613153, 3780295, 4492125, 7295401, 10665331, 11580409, 12511291, 13476375, 15381133
Offset: 1

Views

Author

Amiram Eldar, Sep 25 2019

Keywords

Comments

The prime numbers are excluded from this sequence since tau_p(p) = p for all primes p.
The corresponding values of k are 3, 3, 5, 4, 4, 4, 5, 6, 4, 13, 4, 4, 10, 13, 37, 11, 22, 7, 13, 61, 73, 17, 37, 13, 46, 157, 58, 61, 193, 29, 9, 73, 277, 82, 37, 9, 313, ...

Examples

			18 is in the sequence since tau_3(18) = A007425(18) = 18.

Crossrefs

Cf. A097989.
Cf. A007425, A007426, A034695, A061200, A077592, A111217, A111218, A111219, A111220, A111221, A111306, A163272.

Programs

  • Mathematica
    fun[e_, k_] := Times @@ (Binomial[# + k - 1, k - 1] & /@ e); tau[n_, k_] := fun[ FactorInteger[n][[;; , 2]], k]; aQ[n_] := CompositeQ[n] && Module[{k = 2}, While[(t = tau[n, k]) < n, k++]; t == n]; Select[Range[10^5], aQ]
Previous Showing 41-45 of 45 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.