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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A038993 Number of sublattices of index n in generic 6-dimensional lattice.

Original entry on oeis.org

1, 63, 364, 2667, 3906, 22932, 19608, 97155, 99463, 246078, 177156, 970788, 402234, 1235304, 1421784, 3309747, 1508598, 6266169, 2613660, 10417302, 7137312, 11160828, 6728904, 35364420, 12714681, 25340742, 25095280, 52294536, 21243690, 89572392, 29583456
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Links

Crossrefs

Column 6 of A160870.
Cf. A001001, A038991, A038992, A038994, A038995, A038996, A038997, A038998, A038999.

Programs

  • Mathematica
    f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 5}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

Formula

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=6.
Multiplicative with a(p^e) = product (p^(e+k)-1)/(p^k-1), k=1..5.
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3)*zeta(s-4)*zeta(s-5). Dirichlet convolution of A038992 with A000584. - R. J. Mathar, Mar 31 2011
Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^12*zeta(3)*zeta(5)/3061800 = 0.376266... . - Amiram Eldar, Oct 19 2022

Extensions

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011
More terms from Amiram Eldar, Aug 29 2019

A038995 Number of sublattices of index n in generic 8-dimensional lattice.

Original entry on oeis.org

1, 255, 3280, 43435, 97656, 836400, 960800, 6347715, 8069620, 24902280, 21435888, 142466800, 67977560, 245004000, 320311680, 866251507, 435984840, 2057753100, 943531280, 4241688360, 3151424000, 5466151440, 3559590240, 20820505200, 7947261556, 17334277800, 18326727760
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Links

Crossrefs

Column 8 of A160870.
Cf. A001001, A038991, A038992, A038993, A038994, A038996, A038997, A038998, A038999.

Programs

  • Mathematica
    f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 7}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

Formula

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=8.
Multiplicative with a(p^e) = Product_{k=1..7} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^20*zeta(3)*zeta(5)*zeta(7)/43401015000 = 0.285716... . - Amiram Eldar, Oct 19 2022

Extensions

Offset set to 1 by R. J. Mathar, Mar 01 2011
More terms from Amiram Eldar, Aug 29 2019

A038996 Number of sublattices of index n in generic 9-dimensional lattice.

Original entry on oeis.org

1, 511, 9841, 174251, 488281, 5028751, 6725601, 50955971, 72636421, 249511591, 235794769, 1714804091, 883708281, 3436782111, 4805173321, 13910980083, 7411742281, 37117211131, 17927094321, 85083452531, 66186639441, 120491126959, 81870575521, 501457710611, 198682027181
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Links

Crossrefs

Column 9 of A160870.
Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038997, A038998, A038999.

Programs

  • Mathematica
    f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 8}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

Formula

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=9.
Multiplicative with a(p^e) = Product_{k=1..8} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ c * n^9, where c = Pi^20*zeta(3)*zeta(5)*zeta(7)*zeta(9) / 38578680000 = 0.254479... . - Amiram Eldar, Oct 19 2022

Extensions

Offset changed to 1 by R. J. Mathar, Apr 01 2011
More terms from Amiram Eldar, Aug 29 2019

A038997 Number of sublattices of index n in generic 10-dimensional lattice.

Original entry on oeis.org

1, 1023, 29524, 698027, 2441406, 30203052, 47079208, 408345795, 653757313, 2497558338, 2593742460, 20608549148, 11488207654, 48162029784, 72080070744, 222984027123, 125999618778, 668793731199, 340614792100, 1704167305962, 1389966536992, 2653398536580, 1883023236984
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Links

Crossrefs

Column 10 of A160870.
Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038998, A038999.

Programs

  • Mathematica
    f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 9}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

Formula

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=10.
Multiplicative with a(p^e) = Product_{k=1..9} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ c * n^10, where c = Pi^30*zeta(3)*zeta(5)*zeta(7)*zeta(9) / 4511535509250000 = 0.229259... . - Amiram Eldar, Oct 19 2022

Extensions

Offset set to 1 by R. J. Mathar, Apr 01 2011
More terms from Amiram Eldar, Aug 29 2019

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

A046915 Sum of divisors of 10^n.

Original entry on oeis.org

1, 18, 217, 2340, 24211, 246078, 2480437, 24902280, 249511591, 2497558338, 24987792457, 249938963820, 2499694822171, 24998474116998, 249992370597277, 2499961853010960, 24999809265103951, 249999046325618058, 2499995231628286897
Offset: 0

Views

Author

Enoch Haga

Keywords

Comments

A072692(n) = A049000(n) + a(n).
a(n) is the number of full-dimensional lattices in Z^(n+1) with volume 10. - Álvar Ibeas, Nov 29 2015

Examples

			At 10^1 the factors are 1, 2, 5, 10. The sum of these factors is 18: 1 + 2 + 5 + 10.

Links

Crossrefs

Cf. A000203 (sigma(n)), A049000, A072692. 10th row of A160870, shifted.

Programs

  • Magma
    [1/4*(2^(n+1)-1)*(5^(n+1)-1): n in [0..20]]; // Vincenzo Librandi, Oct 03 2011
  • Mathematica
    Table[DivisorSigma[1, 10^n], {n, 0, 18}] (* Jayanta Basu, Jun 30 2013 *)
  • PARI
    Vec(-(10*x^2-1)/((x-1)*(2*x-1)*(5*x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Jan 27 2015
  • PARI
    a(n) = sigma(10^n); \\ Altug Alkan, Dec 04 2015

Formula

a(n) = 1/4*(2^(n+1)-1)*(5^(n+1)-1). E.g., a(1) = 1/4*(2^2-1)*(5^2-1) = 18. - Vladeta Jovovic, Dec 18 2001
a(n) = 18*a(n-1)-97*a(n-2)+180*a(n-3)-100*a(n-4). - Colin Barker, Jan 27 2015
G.f.: -(10*x^2-1) / ((x-1)*(2*x-1)*(5*x-1)*(10*x-1)). - Colin Barker, Jan 27 2015

A160869 a(n) = sigma(6^(n-1)).

Original entry on oeis.org

1, 12, 91, 600, 3751, 22932, 138811, 836400, 5028751, 30203052, 181308931, 1088123400, 6529545751, 39179682372, 235085301451, 1410533397600, 8463265086751, 50779784492892, 304679288612371, 1828077476115000, 10968470088963751, 65810836228506612
Offset: 1

Views

Author

N. J. A. Sloane, Nov 15 2009

Keywords

Links

Crossrefs

Row 6 of array in A160870.
Cf. A000203, A000400, A059387.

Programs

  • Magma
    [(2^n-1)*(3^n-1)/2: n in [1..50]]; // G. C. Greubel, Apr 30 2018
  • Mathematica
    Table[(2^n-1)*(3^n-1)/2,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{12,-47,72,-36}, {1, 12, 91, 600}, 50] (* G. C. Greubel, Apr 30 2018 *)
  • PARI
    Vec(-x*(6*x^2-1)/((x-1)*(2*x-1)*(3*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Nov 24 2014
  • PARI
    for(n=1, 50, print1((2^n-1)*(3^n-1)/2, ", ")) \\ G. C. Greubel, Apr 30 2018

Formula

a(n) = A059387(n)/2. - Vladimir Joseph Stephan Orlovsky, Apr 28 2010
a(n) = 12*a(n-1)-47*a(n-2)+72*a(n-3)-36*a(n-4). - Colin Barker, Nov 24 2014
G.f.: -x*(6*x^2-1) / ((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Nov 24 2014
a(n) = A000203(A000400(n-1)). - Michel Marcus, Sep 18 2018

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Apr 28 2010
More terms from Colin Barker, Nov 24 2014
Better definition from Altug Alkan, Oct 06 2015

A160891 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 5.

Original entry on oeis.org

1, 15, 40, 120, 156, 600, 400, 960, 1080, 2340, 1464, 4800, 2380, 6000, 6240, 7680, 5220, 16200, 7240, 18720, 16000, 21960, 12720, 38400, 19500, 35700, 29160, 48000, 25260, 93600, 30784, 61440, 58560, 78300, 62400, 129600, 52060, 108600, 95200
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2009

Keywords

Comments

a(n) is the number of lattices L in Z^4 such that the quotient group Z^4 / L is C_nm x (C_m)^3 (and also (C_nm)^3 x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

Examples

			There are 1395 = A160870(8,4) lattices of volume 8 in Z^4. Among them, a(8) = 960 give the quotient group C_8 and a(2) = 15 give C_2 x C_2 x C_2.
Among the lattices of volume 64 in Z^4, there are a(4) = 120 such that the quotient group is C_4 x C_4 x C_4 and other 120 with quotient group C_8 x (C_2)^3.

Links

Crossrefs

Column 4 of A263950.
Cf. A000010, A000203, A059377, A160870.

Programs

  • Maple
    A160891 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(3*e-3)*(1+p+p^2+p^3) ; end do; a; end proc:
seq(A160891(n),n=1..20) ; # R. J. Mathar, Jul 10 2011
  • Mathematica
    A160891[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(5-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 19 2010 *)
f[p_, e_] := p^(3 e - 3)*(1 + p + p^2 + p^3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 08 2022 *)
  • PARI
    vector(50, n, sumdiv(n^3, d, if(ispower(d, 4), moebius(sqrtnint(d, 4))*sigma(n^3/d), 0))) \\ Altug Alkan, Oct 30 2014
  • PARI
    a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(3*f[i,2]-3)*(1+p+p^2+p^3); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

Formula

a(n) = J_4(n)/J_1(n) = J_4(n)/phi(n) = A059377(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 19 2010
Multiplicative with a(p^e) = p^(3e-3)*(1+p+p^2+p^3). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^3). - Álvar Ibeas, Oct 30 2015
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/((p^3-1)*(p^3+p^2+p+1))) = 1.115923965261131974852254388404911045036763705978837384729819264463715993... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.4629765396... . - Amiram Eldar, Nov 08 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^4). - Ridouane Oudra, Apr 01 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Aug 22 2010

A300782 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the simple cubic lattice of index n.

Original entry on oeis.org

1, 3, 3, 9, 5, 13, 7, 24, 14, 23, 11, 49, 15, 33, 31, 66, 21, 70, 25, 89, 49, 61, 33, 162, 50, 81, 75, 137, 49, 177, 55, 193, 97, 123, 99, 296, 75, 147, 129, 312, 89, 291, 97, 269, 218, 203, 113, 534, 146, 302, 203, 357, 141, 451, 207, 508, 247, 307, 171, 789
Offset: 1

Views

Author

Andrey Zabolotskiy, Mar 12 2018

Keywords

Links

Crossrefs

Cf. A159842, A300783, A300784, A003051, A145393, A001001, A128119, A160870, A145396, A145398.

Programs

  • Python
    # see A159842 for the definition of dc, fin, per, u, N, N2
def a(n): # from DeCross's slides
    return (dc(u, N, N2)(n) + 6*dc(fin(1, -1, 0, 4), u, u, N)(n)
      + 3*dc(fin(1, 3), u, u, N)(n)
      + 8*dc(fin(1, 0, -1, 0, 0, 0, 0, 0, 3), u, u, per(0, 1, -1))(n)
      + 6*dc(fin(1, 1), u, u, per(0, 1, 0, -1))(n))//24
print([a(n) for n in range(1, 300)])
# Andrey Zabolotskiy, Sep 02 2019

Extensions

Terms a(11) and beyond from Andrey Zabolotskiy, Sep 02 2019

A362826 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] which commute, divided by n!, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 8, 8, 5, 1, 1, 1, 16, 21, 21, 7, 1, 1, 1, 32, 56, 84, 39, 11, 1, 1, 1, 64, 153, 331, 206, 92, 15, 1, 1, 1, 128, 428, 1300, 1087, 717, 170, 22, 1, 1, 1, 256, 1221, 5111, 5832, 5512, 1810, 360, 30, 1
Offset: 0

Views

Author

Andrew Howroyd, May 09 2023

Keywords

Comments

T(n,k) is also the number of nonisomorphic (k-1)-tuples of permutations of an n-set that pairwise commute. Isomorphism is up to permutation of the elements of the n-set.

Examples

			Array begins:
=======================================================
n/k| 1  2   3    4     5       6        7         8 ...
---+---------------------------------------------------
0  | 1  1   1    1     1       1        1         1 ...
1  | 1  1   1    1     1       1        1         1 ...
2  | 1  2   4    8    16      32       64       128 ...
3  | 1  3   8   21    56     153      428      1221 ...
4  | 1  5  21   84   331    1300     5111     20144 ...
5  | 1  7  39  206  1087    5832    31949    178486 ...
6  | 1 11  92  717  5512   42601   333012   2635637 ...
7  | 1 15 170 1810 19252  208400  2303310  25936170 ...
8  | 1 22 360 5462 81937 1241302 19107225 299002252 ...
  ...

Links

Crossrefs

Columns k=1..4 are A000012, A000041, A061256, A226313.
Cf. A160870, A362827, A362903.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
M(n, m=n)={my(v=vector(m), u=vector(n, n, n==1)); for(j=1, #v, v[j]=concat([1], EulerT(u))~; u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
{ my(A=M(8)); for(n=1, #A~, print(A[n, ])) }

Formula

Column k is the Euler transform of column k-1 of A160870.
T(n,k) = A362827(n,k) / n!.
G.f. of column k: exp(Sum_{i>=1} x^i*A160870(i,k)/i).
G.f. of column k > 1: 1/(Product_{i>=1} (1 - x^i)^A160870(i,k-1)).
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