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 31-37 of 37 results.

A309324 Expansion of Sum_{k>=1} psi(k) * x^k/(1 + x^k), where psi = Dedekind psi function (A001615).

Original entry on oeis.org

1, 2, 5, 2, 7, 10, 9, 2, 17, 14, 13, 10, 15, 18, 35, 2, 19, 34, 21, 14, 45, 26, 25, 10, 37, 30, 53, 18, 31, 70, 33, 2, 65, 38, 63, 34, 39, 42, 75, 14, 43, 90, 45, 26, 119, 50, 49, 10, 65, 74, 95, 30, 55, 106, 91, 18, 105, 62, 61, 70, 63, 66, 153, 2, 105, 130, 69, 38, 125, 126, 73
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 23 2019

Keywords

Comments

Dirichlet convolution of sum of odd divisors function with characteristic function of squarefree numbers.

Links

Crossrefs

Cf. A000593, A001615, A008966, A060648, A193356.

Programs

  • Mathematica
    nmax = 71; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k]  x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[MoebiusMu[n/d]^2 Plus @@ Select[Divisors@ d, OddQ], {d, Divisors[n]}], {n, 1, 71}]
f[2, e_] := 2; f[p_, e_] := (p^e*(p+1)-2)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2, (f[i,1]^f[i,2]*(f[i,1]+1)-2)/(f[i,1]-1)));} \\ Amiram Eldar, Nov 06 2022

Formula

a(n) = Sum_{d|n} (-1)^(n/d+1) * psi(d).
a(n) = Sum_{d|n} mu(n/d)^2 * A000593(d).
Multiplicative with a(2^e) = 2, and a(p^e) = (p^e*(p+1)-2)/(p-1) for odd primes p. - Amiram Eldar, Dec 01 2020
Sum_{k=1..n} a(k) ~ (5/8) * n^2. - Amiram Eldar, Nov 06 2022

A035109 Numerators in the expansion of the Dirichlet series zeta(s) * Product((1+p^-s) / (1-p^(1-s))), p > 2.

Original entry on oeis.org

1, 1, 5, 1, 7, 5, 9, 1, 17, 7, 13, 5, 15, 9, 35, 1, 19, 17, 21, 7, 45, 13, 25, 5, 37, 15, 53, 9, 31, 35, 33, 1, 65, 19, 63, 17, 39, 21, 75, 7, 43, 45, 45, 13, 119, 25, 49, 5, 65, 37, 95, 15, 55, 53, 91, 9, 105, 31, 61, 35, 63, 33, 153, 1, 105, 65, 69, 19, 125, 63, 73, 17, 75, 39
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is also the number of orbits of length n for the map SxT where S has one orbit of each length and T has one orbit of each odd length. - Thomas Ward, Apr 08 2009

Examples

			a(6) = (1/6)*(mu(6)*1*1 + mu(3)*3*1 + mu(2)*4*4 + mu(1)*12*4) = 5. - _Thomas Ward_, Apr 08 2009

Links

Crossrefs

Cf. A008683, A000203, A000593, A001615, A309324, A060648, A000265.

Programs

  • Mathematica
    a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*DivisorSigma[1, #]*DivisorSum[ #, If[OddQ[#], #, 0]&]&]; Array[a, 80] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)
  • PARI
    a(n)=(1/n)*sumdiv(n,d,moebius(n/d)*sigma(d)*sumdiv(d,e,if(e%2,e,0))) \\ Thomas Ward, Apr 08 2009

Formula

Dirichlet g.f.: zeta(s) * Product((1+p^-s) / (1-p^(1-s))), p > 2.
a(n) = (1/n) * Sum_{d|n} mu(n/d) * (Sum_{e|d} e) * (Sum_{e|d, e odd only} e). - Thomas Ward, Apr 08 2009
From Ridouane Oudra, Jun 18 2025: (Start)
a(n) = (1/n) * Sum_{d|n} mu(n/d) * A000203(d) * A000593(d).
a(n) = Sum_{d|n} (psi(2*d) - 2*psi(d)), where psi = A001615.
a(n) = Sum_{d|n, d odd} psi(d).
a(n) = A309324(n) / gcd(n,2).
a(n) = A309324(A000265(n)).
a(n) = A060648(A000265(n)).
a(2*n) = a(n).
a(2*n+1) = A060648(2*n+1). (End)
From Vaclav Kotesovec, Jun 21 2025: (Start)
Dirichlet g.f.: (1 - 2^(1-s)) * zeta(s-1) * zeta(s)^2 / ((1 + 2^(-s)) * zeta(2*s)).
Sum_{k=1..n} a(k) ~ n^2/2. (End)

A062370 a(n) = Sum_{i|n,j|n} sigma(i)*sigma(j)/sigma(gcd(i,j)), where sigma(n) = sum of divisors of n.

Original entry on oeis.org

1, 10, 13, 45, 19, 130, 25, 150, 78, 190, 37, 585, 43, 250, 247, 429, 55, 780, 61, 855, 325, 370, 73, 1950, 174, 430, 358, 1125, 91, 2470, 97, 1122, 481, 550, 475, 3510, 115, 610, 559, 2850, 127, 3250, 133, 1665, 1482, 730, 145, 5577, 310, 1740, 715, 1935
Offset: 1

Views

Author

Vladeta Jovovic, Jul 07 2001

Keywords

Links

Crossrefs

Cf. A000203, A060648, A000005.

Programs

  • Maple
    with(numtheory): seq(add(tau(d^2)*sigma(d), d in divisors(n)), n=1..60); # Ridouane Oudra, Aug 25 2019
  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[0, #^2] * DivisorSigma[1, #] &]; Array[a, 100] (* Amiram Eldar, Sep 15 2019 *)
  • PARI
    a(n) = my(f=factor(n)); for (j=1, #f~, f[j,1] = 1+ sum(k=1, f[j,2], (2*k+1)*sigma(f[j,1]^k)); f[j,2] = 1); factorback(f); \\ Michel Marcus, Feb 28 2019

Formula

Multiplicative with a(p^e) = 1 + Sum_{k=1..e} (2k+1)sigma(p^k). - Mitch Harris, May 24 2005
a(n) = Sum_{d|n} tau(d^2)*sigma(d), where tau(k) = A000005(k) and sigma(k) = A000203(k). - Ridouane Oudra, Aug 25 2019

A344201 Number of cyclic subgroups of the group (C_n)^n, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 4, 14, 136, 782, 23360, 137258, 4210816, 64576643, 2500000768, 28531167062, 2229573502976, 25239592216022, 1852001137606656, 54736740117685528, 2305878194659557376, 51702516367896047762, 6557734713069408616448, 109912203092239643840222
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Links

Crossrefs

Cf. A060648, A064969, A280184, A344219.

Programs

  • Mathematica
    b[n_, k_] := DivisorSum[n, MoebiusMu[n/#] * #^k &] / EulerPhi[n]; a[n_] := DivisorSum[n, b[#, n] &]; Array[a, 20] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    b(n, k) = sumdiv(n, d, moebius(n/d)*d^k)/eulerphi(n);
a(n) = sumdiv(n, d, b(d, n));

Formula

a(n) = Sum_{x_1|n, x_2|n, ... , x_n|n} phi(x_1)*phi(x_2)* ... *phi(x_n)/phi(lcm(x_1, x_2, ... , x_n)).
a(n) = Sum_{d|n} b(d, n), where b(n, k) = ( Sum_{d|n} mu(n/d) * d^k )/phi(n).
If p is prime, a(p) = 1 + (p^p - 1)/(p - 1).

A362624 a(n) = Sum_{d|n, gcd(d,n/d)=1} psi(d), where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 4, 5, 7, 7, 20, 9, 13, 13, 28, 13, 35, 15, 36, 35, 25, 19, 52, 21, 49, 45, 52, 25, 65, 31, 60, 37, 63, 31, 140, 33, 49, 65, 76, 63, 91, 39, 84, 75, 91, 43, 180, 45, 91, 91, 100, 49, 125, 57, 124, 95, 105, 55, 148, 91, 117, 105, 124, 61, 245, 63, 132, 117, 97, 105, 260, 69, 133
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 28 2023

Keywords

Links

Crossrefs

Cf. A001615 (psi), A034444, A060648.

Programs

  • Mathematica
    f[p_, e_] := 1 + (p + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 03 2023 *)

Formula

a(p) = p + 2, p prime.
From Amiram Eldar, May 03 2023: (Start)
Multiplicative with a(p^e) = 1 + (p+1)*p^(e-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1/p^2 + 1/p + p/(1 + p)) = 1.00068765086778318519... . (End)

A362632 a(n) = Sum_{d|n, gcd(d,n/d)=1} d * psi(d), where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 7, 13, 25, 31, 91, 57, 97, 109, 217, 133, 325, 183, 399, 403, 385, 307, 763, 381, 775, 741, 931, 553, 1261, 751, 1281, 973, 1425, 871, 2821, 993, 1537, 1729, 2149, 1767, 2725, 1407, 2667, 2379, 3007, 1723, 5187, 1893, 3325, 3379, 3871, 2257, 5005, 2745, 5257, 3991, 4575, 2863
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 28 2023

Keywords

Links

Crossrefs

Cf. A001615 (psi), A034444, A060648.

Programs

  • Mathematica
    f[p_, e_] := 1 + (p + 1)*p^(2*e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 03 2023 *)

Formula

a(p) = p^2 + p + 1, p prime.
From Amiram Eldar, May 03 2023: (Start)
Multiplicative with a(p^e) = 1 + (p+1)*p^(2*e-1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (p^4 + p^3 + 2*p^2 + 2*p + 1)/(p^2*(p^2 + p + 1)) = 0.55359070186594463118... . (End)

A348011 a(n) = phi(n^2) * Sum_{d|n} 2^omega(d) / d.

Original entry on oeis.org

1, 4, 10, 20, 28, 40, 54, 88, 102, 112, 130, 200, 180, 216, 280, 368, 304, 408, 378, 560, 540, 520, 550, 880, 740, 720, 954, 1080, 868, 1120, 990, 1504, 1300, 1216, 1512, 2040, 1404, 1512, 1800, 2464, 1720, 2160, 1890, 2600, 2856, 2200, 2254, 3680, 2730, 2960
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 24 2021

Keywords

Crossrefs

Cf. A000010, A001221, A002618, A007434, A034444, A047994, A060648, A062355, A069097, A191414, A341772.

Programs

  • Mathematica
    Table[EulerPhi[n^2] DivisorSum[n, 2^PrimeNu[#]/# &], {n, 50}]
f[p_, e_] := p^(e - 1) ((p + 1) p^e - 2); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50]
  • PARI
    a(n) = eulerphi(n^2)*sumdiv(n, d, 2^omega(d)/d); \\ Michel Marcus, Sep 24 2021

Formula

Multiplicative with a(p^e) = p^(e-1) * ((p + 1) * p^e - 2).
a(n) = Sum_{k=1..n, gcd(n,k) = 1} gcd(n,k-1)^2.
a(n) = Sum_{k=1..n} uphi(gcd(n,k)^2).
a(n) = Sum_{d|n} phi(n/d) * uphi(d^2).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.4083249979... . - Amiram Eldar, Nov 05 2022
Previous Showing 31-37 of 37 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.