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.

Showing 1-6 of 6 results.

A160889 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.

Original entry on oeis.org

1, 7, 13, 28, 31, 91, 57, 112, 117, 217, 133, 364, 183, 399, 403, 448, 307, 819, 381, 868, 741, 931, 553, 1456, 775, 1281, 1053, 1596, 871, 2821, 993, 1792, 1729, 2149, 1767, 3276, 1407, 2667, 2379, 3472, 1723, 5187, 1893, 3724, 3627, 3871, 2257, 5824, 2793
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2009

Keywords

Comments

Dirichlet convolution of A000290 and the series of absolute values of A063441. - R. J. Mathar, Jun 20 2011
a(n) is the number of lattices L in Z^3 such that the quotient group Z^3 / L is C_nm x C_m x C_m (and also C_nm x C_nm x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

Examples

			There are 35 = A160870(4,3) lattices of volume 4 in Z^3. Among them, 28 give the quotient group C_4 and 7 give the quotient group C_2 x C_2. Hence, a(4) = 28 and a(2) = 7.
There are 2667 = A160870(32,3) lattices of volume 32 in Z^3. Among them, a(32) = 1792 give the quotient group C_32 (m=1); a(4) = 28 give C_8 x C_2 x C_2 (m=2); a(2) = 7 give C_4 x C_4 x C_2 (m=2).

References

  • J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

Links

Crossrefs

Cf. A156304, A000203.

Programs

  • Mathematica
    A160889[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(4-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Aug 22 2010 *)
  • PARI
    vector(100, n, sumdiv(n^2, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n^2/d), 0))) \\ Altug Alkan, Oct 30 2015

Formula

Moebius transform of A064969. Multiplicative with a(p^e) = (p^2+p+1)*p^(2*e-2). - Vladeta Jovovic, Nov 21 2009
a(n) = J_3(n)/J_1(n)=J_3(n)/phi(n)=A059376(n)/A000010(n), where J_k is the k-th Jordan Totient Function. - Enrique Pérez Herrero, Aug 22 2010
Dirichlet g.f.: zeta(s-2)*product_{primes p} (1+p^(1-s)+p^(-s)). - R. J. Mathar, Jun 20 2011
From Álvar Ibeas, Oct 30 2015: (Start)
a(n) = A254981(n^2). For squarefree n, a(n) = A000203(n^2).
a(n) = Sum_{d|n, n/d squarefree} d^2 * A000203(n/d).
(End)
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.748932997843245303033906997685114802259883493595480897273662144... - Vaclav Kotesovec, Dec 18 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/((p^2-1) * (p^2 + p + 1))) = 1.400940662893945919882073637564538872630336562726971915578687405304250550... - Vaclav Kotesovec, Sep 19 2020
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^3). - Ridouane Oudra, Mar 26 2025

Extensions

Definition corrected by Vladeta Jovovic, Nov 21 2009
Typo in Mathematica program and formula fixed by Enrique Pérez Herrero, Oct 19 2010

A202993 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^4)*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 16, 56, 296, 1052, 4952, 17292, 70512, 249712, 931226, 3212690, 11399590, 38331770, 130310820, 428389292, 1408697596, 4524980036, 14486512316, 45558807176, 142488702483, 439559056419, 1347096766984, 4082169772704, 12286806024269, 36629267989081
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2011

Keywords

Comments

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 16*x^2 + 56*x^3 + 296*x^4 + 1052*x^5 + 4952*x^6 +...
log(A(x)) = x + 31*x^2/2 + 121*x^3/3 + 511*x^4/4 + 781*x^5/5 + 3751*x^6/6 + 2801*x^7/7 + 8191*x^8/8 +...+ A202994(n)*x^n/n +...

Links

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A202994, A156304.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sigma(m^4)*x^m/m)+x*O(x^n)), n)}
  • PARI
    {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k^4)*a(n-k)))}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k^4) * a(n-k) for n>0, with a(0)=1.
log(a(n)) ~ 5 * 3^(1/5) * c^(1/5) * n^(4/5) / 2^(7/5), where c = Product_{primes p} (p*(1 + p + p^2 + p^4) / (p^5 - 1)) = 1.9202928959802946010362130828... - Vaclav Kotesovec, Nov 01 2024

A203557 G.f.: exp( Sum_{n>=1} sigma(n^5)*x^n/n ).

Original entry on oeis.org

1, 1, 32, 153, 1145, 5677, 37641, 184685, 1047862, 5196410, 26935148, 129702476, 638028933, 2987297287, 14055935617, 64139004752, 291595380989, 1296984485909, 5732084828019, 24910785830408, 107411267744602, 457008372687439, 1928413165110846, 8046605441623654
Offset: 0

Views

Author

Paul D. Hanna, Jan 03 2012

Keywords

Comments

In general, if m >= 1 and g.f.= exp(Sum_{k>=1} sigma(k^m)*x^k/k), then log(a(n)) ~ (1 + 1/m) * (c*m!)^(1/(m+1)) * n^(m/(m+1)), where c = Product_{primes p} ((p^(m+2) - p^(m+1) + p^m - p) / ((p-1)*(p^(m+1)-1))). - Vaclav Kotesovec, Nov 01 2024

Examples

			G.f.: A(x) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...
where the logarithm equals the l.g.f. of A203556:
log(A(x)) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...+ sigma(n^5)*x^n/n +...

Links

Crossrefs

Cf. A203556, A000203 (sigma); variants: A000041 (m=1), A156303 (m=2), A156304 (m=3), A202993 (m=4).

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,sigma(m^5)*x^m/m)+x*O(x^n)),n)}

Formula

Logarithmic derivative yields A203556.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A203556(k)*a(n-k) for n > 0. - Seiichi Manyama, Sep 09 2020
log(a(n)) ~ 2^(3/2) * 3^(7/6) * c^(1/6) * n^(5/6) / 5^(5/6), where c = Product_{primes p} (p*(1 + p + p^2 + p^3 + p^5) / (p^6 - 1)) = 1.93252811194652723494722635658171746713... - Vaclav Kotesovec, Nov 01 2024

A319363 a(n) = [x^n] exp(Sum_{k>=1} sigma(k^n)*x^k/k).

Original entry on oeis.org

1, 1, 4, 21, 296, 5677, 291348, 22679763, 3946629792, 1281287791090, 840017139222515, 1068253745514088673, 2745322115165570881474, 14006438682727577999625359, 141884380264686466724199980066, 2897075017978846591982107951045864, 118770312781918226439371316982748736112
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2018

Keywords

Links

Crossrefs

Cf. A000041, A156303, A156304, A202993, A203557, A319362.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[DivisorSigma[1, k^n] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

A381712 Euler transform of n * A194532(n).

Original entry on oeis.org

1, 1, 43, 316, 2563, 17284, 135843, 903141, 6153645, 39839122, 256023118, 1589382754, 9751548710, 58451287319, 345478493273, 2006641555356, 11498560570683, 64940715401160, 362249937059777, 1995639600211016, 10870475203155005, 58563229198239242, 312277069694594537
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2025

Keywords

Crossrefs

Cf. A156304, A381714.
Cf. A194532.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^3, 2)*x^k/k)))

Formula

G.f.: 1/Product_{k>=1} (1 - x^k)^(k * A194532(k)).
G.f.: exp( Sum_{k>=1} sigma_2(k^3) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_2(k^3) * a(n-k).

A381714 Euler transform of n^2 * A381713(n).

Original entry on oeis.org

1, 1, 293, 7106, 124636, 2507807, 53728922, 975224769, 17336813339, 308906655193, 5324331825516, 88599795614719, 1449812221707335, 23313054134280890, 367282089624429463, 5682414281863178845, 86571519001530856417, 1299264182863131989813
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2025

Keywords

Crossrefs

Cf. A156304, A381712.
Cf. A381713.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^3, 3)*x^k/k)))

Formula

G.f.: 1/Product_{k>=1} (1 - x^k)^(k^2 * A381713(k)).
G.f.: exp( Sum_{k>=1} sigma_3(k^3) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_3(k^3) * a(n-k).
Showing 1-6 of 6 results.

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

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