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 1-10 of 11 results. Next

A180608 O.g.f.: exp( Sum_{n>=1} A067692(n)*x^n/n ), where A067692(n) = [sigma(n)^2 + sigma(n,2)]/2.

Original entry on oeis.org

1, 1, 4, 8, 21, 39, 93, 171, 364, 675, 1338, 2433, 4641, 8282, 15222, 26811, 47920, 83046, 145288, 248164, 425970, 718303, 1213106, 2020540, 3365352, 5541996, 9115640, 14856657, 24164430, 39002462, 62800603, 100454208, 160257140
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2010

Keywords

Comments

sigma(n) = A000203(n), sum of divisors of n;
sigma(n,2) = A001157(n), sum of squares of divisors of n.

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 93*x^6 +...
log(A(x)) = x + 7*x^2/2 + 13*x^3/3 + 35*x^4/4 + 31*x^5/5 + 97*x^6/6 +...

Links

Crossrefs

Cf. A067692, A000203, A001157, A072861, A156302.

Programs

  • Mathematica
    nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[(DivisorSigma[1, k]^2 + DivisorSigma[2, k])/2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 28 2024 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, (sigma(m)^2+sigma(m,2))/2*x^m/m)+x*O(x^n)), n)}

Formula

log(a(n)) ~ 3*(7*zeta(3))^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Oct 29 2024

A068020 a(n) = Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=3.

Original entry on oeis.org

1, 15, 40, 155, 156, 672, 400, 1395, 1210, 2520, 1464, 7280, 2380, 6336, 6600, 11811, 5220, 21030, 7240, 26880, 16672, 22752, 12720, 66960, 20306, 36792, 33880, 67040, 25260, 119592, 30784, 97155, 60144, 80136, 64080, 230966, 52060, 110880, 97384
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Links

Crossrefs

Cf. A067692, A068021, A068022, A068023, A068024, A068025, A068026, A068027.
Cf. A000203, A001157, A001158.

Programs

  • Mathematica
    a[n_] := 1/3!*(DivisorSigma[1, n]^3 + 3*DivisorSigma[1, n]*DivisorSigma[2, n] + 2*DivisorSigma[3, n]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Dec 12 2011, after given formula *)
CIP3 = CycleIndexPolynomial[SymmetricGroup[3], Array[x, 3]]; a[n_] := CIP3 /. x[k_] -> DivisorSigma[k, n]; Array[a, 39] (* Jean-François Alcover, Nov 04 2016 *)
  • PARI
    a(n) = my(f = factor(n)); (2*sigma(f, 3) + 3*sigma(f, 1)*sigma(f, 2) + sigma(f)^3) / 6; \\ Amiram Eldar, Jan 03 2025

Formula

a(n) = (1/3!)*(sigma_1(n)^3 + 3*sigma_1(n)*sigma_2(n) + 2*sigma_3(n)).
a(n) = Sum_{r|n, s|n, t|n, r<=s<=t} r*s*t.
From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: (zeta(s)*zeta(s-3)/6) * (zeta(s-1)*zeta(s-2) * (f(s) + 3/zeta(2*s-3)) + 2), where f(s) = Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (7/96) * zeta(3) * zeta(6) * Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) + zeta(2)*zeta(3)*zeta(4)/(8*zeta(5)) + zeta(4)/12 = 0.60106209766277728837... . (End)

A068027 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=10.

Original entry on oeis.org

1, 2047, 88573, 2794155, 12207031, 217414561, 329554457, 3269560515, 5883904390, 27757433067, 28531167061, 323579719281, 149346699503, 726383654349, 1158434219878, 3571013994483, 2141993519227, 15288525998824
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068026, A000203, A001157-A001160, A013954-A013958.

Programs

  • Mathematica
    CIP10 = CycleIndexPolynomial[SymmetricGroup[10], Array[x, 10]]; a[n_] := CIP10 /. x[k_] -> DivisorSigma[k, n]; Array[a, 18] (* Jean-François Alcover, Nov 04 2016 *)

A119616 Second elementary symmetric function of divisors of n.

Original entry on oeis.org

0, 2, 3, 14, 5, 47, 7, 70, 39, 97, 11, 287, 13, 163, 158, 310, 17, 533, 19, 609, 262, 343, 23, 1375, 155, 457, 390, 1043, 29, 1942, 31, 1302, 542, 733, 502, 3185, 37, 895, 718, 2945, 41, 3358, 43, 2247, 1859, 1267, 47, 5983, 399, 2697, 1142, 3017, 53, 5150, 1006
Offset: 1

Views

Author

N. J. A. Sloane, based on email from Neven Juric (neven.juric(AT)apis-it.hr), Jun 07 2006

Keywords

Comments

a(p)=p if p is prime and records are A002093 (highly abundant numbers). - Robert G. Wilson v, Jun 07 2006

Examples

			|-------+------------------------------------------+---------------------|
|...n...|................divisors(n)...............|..s2(divisors.(n))...|
|-------+------------------------------------------+---------------------|
|...1...|....................1.....................|..........0..........|
|...2...|...................1,2....................|..........2..........|
|...3...|...................1,3....................|..........3..........|
|...4...|..................1,2,4...................|.........14..........|
|...5...|...................1,5....................|..........5..........|
|...6...|.................1,2,3,6..................|.........47..........|

Links

Crossrefs

Cf. A002093, A000203, A001157, A002117, A067692.
Column k=2 of A224381.

Programs

  • Maple
    a:= n-> (l-> add(add(l[i]*l[j], i=1..j-1), j=2..nops(l)))
        (sort([numtheory[divisors](n)[]])):
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 25 2014
  • Mathematica
    f[n_] := Block[{d = Divisors@n}, Sum[ d[[u]]*d[[v]], {v, 2, Length@d}, {u, v - 1}]]; Array[f, 55] (* Robert G. Wilson v *)
  • PARI
    a(n)=my(d=divisors(n));sum(i=1,#d-1,sum(j=i+1,#d,d[i]*d[j])) \\ Charles R Greathouse IV, Mar 05 2013
  • PARI
    a(n)=(sigma(n)^2-sigma(n,2))/2 \\ Charles R Greathouse IV, Mar 05 2013

Formula

a(n) = Sum_{u|n, v|n, u
a(n) = (sigma_1(n)^2-sigma_2(n))/2, cf. A000203, A001157. - Vladeta Jovovic, Jun 07 2006
Sum_{k=1..n} a(k) = zeta(3) * n^3 / 4 + O(n^2*log(n)^2). - Amiram Eldar, Dec 15 2023

A068022 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=5.

Original entry on oeis.org

1, 63, 364, 2667, 3906, 26964, 19608, 97155, 99463, 271278, 177156, 1228836, 402234, 1324008, 1520784, 3309747, 1508598, 7746453, 2613660, 12021702, 7487424, 11661372, 6728904, 46371780, 12714681, 26297334, 25095280, 57926792
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020, A068021, A068023-A068027, A000203, A001157-A001160.

Programs

  • Mathematica
    CIP5 = CycleIndexPolynomial[SymmetricGroup[5], Array[x, 5]]; a[n_] := CIP5 /. x[k_] -> DivisorSigma[k, n]; Array[a, 28] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/5!*(sigma[1](n)^5 + 10*sigma[1](n)^3*sigma[2](n) + 20*sigma[1](n)^2*sigma[3](n) + 15*sigma[1](n)*sigma[2](n)^2 + 30*sigma[1](n)*sigma[4](n) + 20*sigma[2](n)*sigma[3](n) + 24*sigma[5](n)).

A068023 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=6.

Original entry on oeis.org

1, 127, 1093, 10795, 19531, 164809, 137257, 788035, 896260, 2745247, 1948717, 15172249, 5229043, 18728221, 22858948, 53743987, 25646167, 142560946, 49659541, 244930015, 157475284, 258931921, 154764793, 1151073625, 317886556
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068022, A068024-A068027, A000203, A001157-A001160, A013954.

Programs

  • Mathematica
    CIP6 = CycleIndexPolynomial[SymmetricGroup[6], Array[x, 6]]; a[n_] := CIP6 /. x[k_] -> DivisorSigma[k, n]; Array[a, 25] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/6!*(sigma[1](n)^6 + 15*sigma[1](n)^4*sigma[2](n) + 40*sigma[1](n)^3*sigma[3](n) + 45*sigma[1](n)^2*sigma[2](n)^2 + 90*sigma[1](n)^2*sigma[4](n) + 120*sigma[1](n)*sigma[2](n)*sigma[3](n) + 15*sigma[2](n)^3 + 144*sigma[1](n)*sigma[5](n) + 90*sigma[2](n)*sigma[4](n) + 40*sigma[3](n)^2 + 120*sigma[6](n)).
Agrees with A038994 at n = 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23... - Ralf Stephan, Mar 09 2004

A068024 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=7.

Original entry on oeis.org

1, 255, 3280, 43435, 97656, 998184, 960800, 6347715, 8069620, 27615060, 21435888, 184770040, 67977560, 263540112, 343123440, 866251507, 435984840, 2595218340, 943531280, 4944199260, 3308659904, 5722701624, 3559590240
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068023, A068025-A068027, A000203, A001157-A001160, A013954, A013955.

Programs

  • Mathematica
    CIP7 = CycleIndexPolynomial[SymmetricGroup[7], Array[x, 7]]; a[n_] := CIP7 /. x[k_] -> DivisorSigma[k, n]; Array[a, 23] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/7!*(sigma[1](n)^7 + 21*sigma[1](n)^5*sigma[2](n) + 70*sigma[1](n)^4*sigma[3](n) + 105*sigma[1](n)^3*sigma[2](n)^2 + 210*sigma[1](n)^3*sigma[4](n) + 420*sigma[1](n)^2*sigma[2](n)*sigma[3](n) + 105*sigma[1](n)*sigma[2](n)^3 + 504*sigma[1](n)^2*sigma[5](n) + 630*sigma[1](n)*sigma[2](n)*sigma[4](n) + 280*sigma[1](n)*sigma[3](n)^2 + 210*sigma[2](n)^2*sigma[3](n) + 840*sigma[1](n)*sigma[6](n) + 504*sigma[2](n)*sigma[5](n) + 420*sigma[3](n)*sigma[4](n) + 720*sigma[7](n)).

A068025 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=8.

Original entry on oeis.org

1, 511, 9841, 174251, 488281, 6017605, 6725601, 50955971, 72636421, 276964061, 235794769, 2234070293, 883708281, 3698977205, 5148057541, 13910980083, 7411742281, 46982039533, 17927094321, 99343345101, 69493620405
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068024, A068026-A068027, A000203, A001157-A001160, A013954-A013956.

Programs

  • Mathematica
    CIP8 = CycleIndexPolynomial[SymmetricGroup[8], Array[x, 8]]; a[n_] := CIP8 /. x[k_] -> DivisorSigma[k, n]; Array[a, 21] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/8!*(sigma[1](n)^8 + 28*sigma[1](n)^6*sigma[2](n) + 112*sigma[1](n)^5*sigma[3](n) + 210*sigma[1](n)^4*sigma[2](n)^2 + 420*sigma[1](n)^4*sigma[4](n) + 1120*sigma[1](n)^3*sigma[2](n)*sigma[3](n) + 420*sigma[1](n)^2*sigma[2](n)^3 + 1344*sigma[1](n)^3*sigma[5](n) + 2520*sigma[1](n)^2*sigma[2](n)*sigma[4](n) + 1120*sigma[1](n)^2*sigma[3](n)^2 + 1680*sigma[1](n)*sigma[2](n)^2*sigma[3](n) + 105*sigma[2](n)^4 + 3360*sigma[1](n)^2*sigma[6](n) + 4032*sigma[1](n)*sigma[2](n)*sigma[5](n) + 3360*sigma[1](n)*sigma[3](n)*sigma[4](n) + 1260*sigma[2](n)^2*sigma[4](n) + 1120*sigma[2](n)*sigma[3](n)^2 + 5760*sigma[7](n)*sigma[1](n) + 3360*sigma[2](n)*sigma[6](n) + 2688*sigma[3](n)*sigma[5](n) + 1260*sigma[4](n)^2 + 5040*sigma[8](n)).

A068026 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=9.

Original entry on oeis.org

1, 1023, 29524, 698027, 2441406, 36192156, 47079208, 408345795, 653757313, 2773708938, 2593742460, 26912354924, 11488207654, 51851591352, 77226922344, 222984027123, 125999618778, 848125888467, 340614792100, 1991478050562
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068025, A068027, A000203, A001157-A001160, A013954-A013957.

Programs

  • Mathematica
    CIP9 = CycleIndexPolynomial[SymmetricGroup[9], Array[x, 9]]; a[n_] := CIP9 /. x[k_] -> DivisorSigma[k, n]; Array[a, 20] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/9!*(sigma[1](n)^9 + 36*sigma[1](n)^7*sigma[2](n) + 168*sigma[1](n)^6*sigma[3](n) + 378*sigma[1](n)^5*sigma[2](n)^2 + 756*sigma[1](n)^5*sigma[4](n) + 2520*sigma[1](n)^4*sigma[2](n)*sigma[3](n) +
+ 1260*sigma[1](n)^3*sigma[2](n)^3 + 3024*sigma[1](n)^4*sigma[5](n) + 7560*sigma[1](n)^3*sigma[2](n)*sigma[4](n) + 3360*sigma[1](n)^3*sigma[3](n)^2 + 7560*sigma[1](n)^2*sigma[2](n)^2*sigma[3](n) +
+ 945*sigma[1](n)*sigma[2](n)^4 + 10080*sigma[1](n)^3*sigma[6](n) + 18144*sigma[1](n)^2*sigma[2](n)*sigma[5](n) + 15120*sigma[1](n)^2*sigma[3](n)*sigma[4](n) + 11340*sigma[1](n)*sigma[2](n)^2*sigma[4](n) + 10080*sigma[1](n)*sigma[2](n)*sigma[3](n)^2 + 2520*sigma[2](n)^3*sigma[3](n) + 25920*sigma[7](n)*sigma[1](n)^2 +
+ 30240*sigma[1](n)*sigma[2](n)*sigma[6](n) + 24192*sigma[1](n)*sigma[3](n)*sigma[5](n) + 11340*sigma[1](n)*sigma[4](n)^2 + 9072*sigma[2](n)^2*sigma[5](n) + 15120*sigma[2](n)*sigma[3](n)*sigma[4](n) + 2240*sigma[3](n)^3 + 25920*sigma[7](n)*sigma[2](n) + 45360*sigma[8](n)*sigma[1](n) + 20160*sigma[3](n)*sigma[6](n) + 18144*sigma[4](n)*sigma[5](n) + 40320*sigma[9](n)).

A067817 a(n) = Sum_{r|n, s|n, t|n, r

Original entry on oeis.org

0, 0, 0, 8, 0, 72, 0, 120, 27, 180, 0, 1400, 0, 336, 360, 1240, 0, 3285, 0, 3948, 672, 792, 0, 15960, 125, 1092, 1080, 8240, 0, 25992, 0, 11160, 1584, 1836, 1680, 57065, 0, 2280, 2184, 46620, 0, 56352, 0, 23592, 18612, 3312, 0, 150040, 343, 29955, 3672
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Links

Crossrefs

Cf. A067692, A068020, A000203, A001157, A001158.
Column k=3 of A224381.

Programs

  • Maple
    a:= n-> coeff(expand(mul(1+d*x, d=numtheory[divisors](n))), x, 3):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 18 2023
  • Mathematica
    a[n_] := Module[{d = DivisorSigma[{1, 2, 3}, n]}, (d[[1]]^3 - 3*d[[1]]*d[[2]] + 2*d[[3]]) / 6]; Array[a, 50] (* Amiram Eldar, Jan 03 2025 *)
  • PARI
    a(n) = 1/6*(sigma(n, 1)^3 - 3*sigma(n, 1)*sigma(n, 2) + 2*sigma(n, 3)) \\ Michel Marcus, Jun 17 2013

Formula

a(n) = (1/3!)*(sigma_1(n)^3 - 3*sigma_1(n)*sigma_2(n) + 2*sigma_3(n)).
From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: (zeta(s)*zeta(s-3)/6) * (zeta(s-1)*zeta(s-2) * (f(s) - 3/zeta(2*s-3)) + 2), where f(s) = Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (7/96) * zeta(3) * zeta(6) * Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) - zeta(2)*zeta(3)*zeta(4)/(8*zeta(5)) + zeta(4)/12 = 0.085094994884972381542... . (End)
Showing 1-10 of 11 results. Next

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

Content is available under The OEIS End-User License Agreement.