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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A344043 a(n) = n * Sum_{d|n} sigma(d)^3 / d.

Original entry on oeis.org

1, 29, 67, 401, 221, 1943, 519, 4177, 2398, 6409, 1739, 26867, 2757, 15051, 14807, 38145, 5849, 69542, 8019, 88621, 34773, 50431, 13847, 279859, 30896, 79953, 71194, 208119, 27029, 429403, 32799, 326337, 116513, 169621, 114699, 961598, 54909, 232551, 184719, 923117, 74129, 1008417, 85227, 697339
Offset: 1

Views

Author

Seiichi Manyama, May 08 2021

Keywords

Links

Crossrefs

Cf. A002117, A060640, A226565, A226566, A344044.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, DivisorSigma[1, #]^3/# &]; Array[a, 44] (* Amiram Eldar, May 08 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, sigma(d)^3/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^3*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k >= 1} sigma(k)^3 * x^k/(1 - x^k)^2.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (Pi^6*zeta(3)^2/2160) * Product_{p prime} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 1.8238925519... . - Amiram Eldar, Nov 20 2022

A349123 a(n) = Sum_{d|n} A038040(n/d) * A003415(d), where A038040(n) = n*tau(n), and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 8, 1, 15, 1, 40, 12, 21, 1, 96, 1, 27, 24, 160, 1, 126, 1, 144, 30, 39, 1, 440, 20, 45, 90, 192, 1, 279, 1, 560, 42, 57, 36, 720, 1, 63, 48, 680, 1, 369, 1, 288, 234, 75, 1, 1680, 28, 270, 60, 336, 1, 810, 48, 920, 66, 93, 1, 1656, 1, 99, 306, 1792, 54, 549, 1, 432, 78, 531, 1, 3120, 1, 117, 330, 480, 54, 639
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

This sequence is the Dirichlet convolution of at least the following pairs of sequences:
- A003415 (the arithmetic derivative) with A038040,
- A000027 (the identity function) with A347130,
- A000203 (sigma) with A347131,
- A018804 with A319684,
- A060640 with A300251.

Links

Crossrefs

Cf. A000027, A000203, A003415, A003557, A018804, A060640, A300251, A319684, A347130, A349124.
Cf. also A348983, A349143.

Programs

  • Mathematica
    d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A038040(n) = (n*numdiv(n));
A349123(n) = sumdiv(n,d,A038040(d)*A003415(n/d));

Formula

a(n) = Sum_{d|n} A038040(d) * A003415(n/d).
a(n) = Sum_{d|n} d * A347130(n/d).
a(n) = Sum_{d|n} A000203(d) * A347131(n/d).
a(n) = Sum_{d|n} A018804(d) * A319684(n/d).
a(n) = Sum_{d|n} A060640(d) * A300251(n/d).
For all n >= 1, A348983(n) <= a(n) <= A349143(n).
a(n) = A003557(n) * A349124(n).

A027844 Number of subgroups of index n of fundamental group of the non-orientable cycle bundle over the Klein bottle.

Original entry on oeis.org

1, 7, 7, 27, 11, 55, 15, 91, 34, 97, 23, 231, 27, 147, 77, 299, 35, 334, 39, 437, 105, 271, 47, 847, 86, 345, 142, 699, 59, 865, 63, 1003, 161, 517, 165, 1590, 75, 615, 189, 1701, 83, 1371, 87, 1391, 374, 835, 95, 3023, 162, 1322, 245, 1821, 107, 2062, 253, 2835
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From a recent general formula of Stanley's for the number of subgroups in G\times Z.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64.

Links

Crossrefs

Cf. A001001, A027845, A046524.

Programs

  • Mathematica
    b[k_] := If[OddQ[k], DivisorSigma[0, k], (3 DivisorSigma[0, k] + DivisorSigma[1, k/2] - DivisorSigma[0, k/2])/2]; a[n_] := Sum[k*b[k], {k, Divisors[n]}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jul 19 2012 *)
  • PARI
    A001001(n) = sumdiv(n, d, sigma(d) * d);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
a(n) = S1(n) + S11(n) + S21(n);
vector(56, n, a(n))  \\ Gheorghe Coserea, May 05 2016

Formula

Sum k*b(k), k|n, where b(k) is the number of n-list coverings of the Klein bottle (A046524).

Extensions

More terms from Valery A. Liskovets
Corrected and extended by Vladeta Jovovic, Feb 03 2003

A061259 a(n)=Sum_{d|n} d*numbpart(d), where numbpart(d)=number of partitions of d, cf. A000041.

Original entry on oeis.org

1, 5, 10, 25, 36, 80, 106, 201, 280, 460, 617, 1024, 1314, 2000, 2685, 3897, 5050, 7280, 9311, 13020, 16747, 22665, 28866, 39000, 48986, 64654, 81550, 106124, 132386, 171295, 212103, 271065, 335345, 423594, 521046, 655396, 800570, 997885
Offset: 1

Views

Author

Vladeta Jovovic, Apr 21 2001

Keywords

Links

Crossrefs

Cf. A000041, A057660, A001001, A060640, A061258.
Cf. A001970, A027750.

Programs

  • Haskell
    a061259 n = sum $ zipWith (*) divs $ map a000041 divs
            where divs = a027750_row' n
-- Reinhard Zumkeller, Oct 31 2015

A327960 Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1)^2).

Original entry on oeis.org

1, -5, -7, 8, -11, 35, -15, -4, 15, 55, -23, -56, -27, 75, 77, 0, -35, -75, -39, -88, 105, 115, -47, 28, 35, 135, -9, -120, -59, -385, -63, 0, 161, 175, 165, 120, -75, 195, 189, 44, -83, -525, -87, -184, -165, 235, -95, 0, 63, -175, 245, -216, -107, 45, 253, 60, 273, 295, -119, 616
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 22 2019

Keywords

Comments

Dirichlet inverse of A060640.
Moebius transform applied twice to A101035.

Links

Crossrefs

Cf. A046101 (positions of 0's), A046692, A055615, A060640, A101035.

Programs

  • Mathematica
    a[1] = 1; a[n_] := -Sum[Sum[j DivisorSigma[0, j], {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 60}]
f[p_, e_] := Which[e==1, -(2*p+1), e==2, p^2+2*p, e==3, -p^2, e>3, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA060640(n/d) * a(d).
a(n) = Sum_{d|n} A046692(n/d) * A055615(d).
Multiplicative with a(p^e) = -(2*p+1) if e=1, p^2+2*p if e=2, -p^2 if e=3, and 0 otherwise. - Amiram Eldar, Dec 02 2020

A343525 If n = Product (p_j^k_j) then a(n) = Product (2*p_j^k_j + 1), with a(1) = 1.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 17, 19, 55, 23, 63, 27, 75, 77, 33, 35, 95, 39, 99, 105, 115, 47, 119, 51, 135, 55, 135, 59, 385, 63, 65, 161, 175, 165, 171, 75, 195, 189, 187, 83, 525, 87, 207, 209, 235, 95, 231, 99, 255, 245, 243, 107, 275, 253, 255, 273, 295, 119, 693, 123, 315, 285, 129, 297, 805
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 18 2021

Keywords

Comments

The unitary analog of A060640.

Links

Crossrefs

Cf. A001221, A034444, A034448, A060640, A107759, A145388, A298473.

Programs

  • Maple
    a:= n-> mul(2*i[1]^i[2]+1, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 18 2021
  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((2 #[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 66}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2*f[k,1]^f[k,2]+1; f[k,2]=1); factorback(f); \\ Michel Marcus, Apr 18 2021

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * usigma(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-1) - 2/p^(2*s-1)). - Amiram Eldar, Jul 24 2024

A263829 Total number c_{pi_1(B_2)}(n) of n-coverings over the second amphicosm.

Original entry on oeis.org

1, 3, 5, 13, 7, 19, 9, 43, 18, 33, 13, 93, 15, 51, 35, 137, 19, 110, 21, 175, 45, 99, 25, 355, 38, 129, 58, 285, 31, 289, 33, 455, 65, 201, 63, 626, 39, 243, 75, 721, 43, 483, 45, 589, 126, 339, 49, 1305, 66, 498, 95, 783, 55, 750, 91, 1227
Offset: 1

Views

Author

N. J. A. Sloane, Oct 28 2015

Keywords

Links

Crossrefs

Cf. A263825-A263830, A263832.

Programs

  • PARI
    A001001(n) = sumdiv(n, d, sigma(d) * d);
A007429(n) = sumdiv(n, d, sigma(d));
A007434(n) = sumdiv(n, d, moebius(n\d) * d^2);
A059376(n) = sumdiv(n, d, moebius(n\d) * d^3);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
EpiPcZn(n) = sumdiv(n, d, moebius(n\d) * d^2 * gcd(d,2));
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
S22(n)     = { if (n%2, A060640(n), if (n%4, 0,
  sumdiv(n\4, d, 2*d*(sigma(n\(2*d)) - sigma(n\(4*d))))));
};
A027844(n) = S1(n) + S11(n) + S21(n);
a(n) = { 1/n * sumdiv(n, d,
  A059376(d) * S1(n\d) + EpiPcZn(d) * S21(n\d) + A007434(d) * S22(n\d));
};
vector(56, n, a(n))  \\ Gheorghe Coserea, May 04 2016

Extensions

More terms from Gheorghe Coserea, May 04 2016

A307793 a(1) = 1; a(n+1) = Sum_{d|n} tau(d)*a(d), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 1, 3, 7, 24, 49, 205, 411, 1668, 5011, 20095, 40191, 241372, 482745, 1931393, 7725627, 38629803, 77259607, 463562851, 927125703, 5562774334, 22251097753, 89004431205, 178008862411, 1424071142304, 4272213426961, 17088854190591, 68355416767375, 410132502535664, 820265005071329
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 29 2019

Keywords

Crossrefs

Cf. A000005, A007557, A060640, A307794, A319133.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[DivisorSigma[0, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 30}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[DivisorSigma[0, k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 30}]
  • PARI
    a(n) = if (n==1, 1, sumdiv(n-1, d, numdiv(d)*a(d))); \\ Michel Marcus, Apr 29 2019

Formula

G.f.: x * (1 + Sum_{n>=1} tau(n)*a(n)*x^n/(1 - x^n)).
L.g.f.: -log(Product_{i>=1, j>=1} (1 - x^(i*j))^(a(i*j)/(i*j))) = Sum_{n>=1} a(n+1)*x^n/n.

A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

Original entry on oeis.org

1, 3, 5, 7, 7, 15, 9, 15, 18, 21, 13, 35, 15, 27, 35, 31, 19, 54, 21, 49, 45, 39, 25, 75, 38, 45, 58, 63, 31, 105, 33, 63, 65, 57, 63, 126, 39, 63, 75, 105, 43, 135, 45, 91, 126, 75, 49, 155, 66, 114, 95, 105, 55, 174, 91, 135, 105, 93, 61, 245, 63, 99
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 13 2019

Keywords

Comments

Inverse Moebius transform of A002131.
Dirichlet convolution of A000027 with A001227.

Links

Crossrefs

Cf. A000203, A001227, A002131, A007429, A026741, A060640, A109386, A133700, A288417, A300707.

Programs

  • Mathematica
    nmax = 62; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[Total[Select[Divisors[d], OddQ[d/#] &]], {d, Divisors[n]}]; Table[a[n], {n, 1, 62}]
  • PARI
    a(n)={sumdiv(n, d, if(n/d%2, sigma(d)))} \\ Andrew Howroyd, Sep 13 2019

Formula

G.f.: Sum_{k>=1} A002131(k) * x^k / (1 - x^k).
G.f.: Sum_{k>=1} A001227(k) * x^k / (1 - x^k)^2.
a(n) = Sum_{d|n} A002131(d).
a(n) = Sum_{d|n} d * A001227(n/d).
a(n) = (A007429(n) + A288417(n)) / 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). - Amiram Eldar, Oct 23 2022

A069914 a(n) = Sum_{d|n} (d-1)*sigma(n/d).

Original entry on oeis.org

0, 1, 2, 6, 4, 15, 6, 23, 16, 27, 10, 64, 12, 39, 42, 72, 16, 98, 18, 110, 60, 63, 22, 213, 48, 75, 84, 156, 28, 245, 30, 201, 96, 99, 102, 380, 36, 111, 114, 357, 40, 345, 42, 248, 248, 135, 46, 618, 96, 278, 150, 294, 52, 478, 162, 501, 168, 171, 58, 924, 60, 183
Offset: 1

Views

Author

Vladeta Jovovic, May 04 2002

Keywords

Links

Crossrefs

Cf. A000203, A006579, A007429, A060640.

Programs

  • Mathematica
    Table[Plus @@ (DivisorSigma[1, ds = Divisors[n]]*(n/ds - 1)), {n, 62}] (* Ivan Neretin, May 17 2015 *)
  • PARI
    a(n) = sumdiv(n, d, (d-1)*sigma(n/d)) \\ Michel Marcus, Jun 17 2013

Formula

a(n) = A060640(n) - A007429(n).
G.f.: Sum_{k>=1} sigma(k) * x^(2*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Aug 19 2021
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