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 11-15 of 15 results.

A161010 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 14.

Original entry on oeis.org

1, 8191, 797161, 33550336, 305175781, 6529545751, 16148168401, 137422176256, 423644039001, 2499694822171, 3452271214393, 26745019396096, 25239592216021, 132269647372591, 243274230757741, 562881233944576, 619036127056621
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2009

Keywords

Comments

a(n) is the number of lattices L in Z^13 such that the quotient group Z^13 / L is C_n. - Álvar Ibeas, Nov 26 2015

Links

Crossrefs

Column 13 of A263950.
Cf. A000010, A013670, A000203, A013671, A160897.

Programs

  • Maple
    f:= proc(n) local t; mul(t[1]^(12*t[2]-12)*(t[1]^13-1)/(t[1]-1), t = ifactors(n)[2]) end proc:
seq(f(n),n=1..100); # Robert Israel, Dec 08 2015
  • Mathematica
    b = 14; Table[Sum[MoebiusMu[n/d] d^(b - 1), {d, Divisors@ n}]/EulerPhi@ n, {n, 17}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(12*e - 12) * (p^13-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
  • PARI
    vector(100, n, sumdiv(n^12, d, if(ispower(d, 13), moebius(sqrtnint(d, 13))*sigma(n^12/d), 0))) \\ Altug Alkan, Nov 26 2015
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^13 - 1)*f[i,1]^(12*f[i,2] - 12)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

Formula

a(n) = J_13(n)/J_1(n) where J_13 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(12e-12) * (p^13-1) / (p-1).
For squarefree n, a(n) = A000203(n^12). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^13, where c = (1/13) * Product_{p prime} (1 + (p^12-1)/((p-1)*p^13)) = 0.14949521105... .
Sum_{k>=1} 1/a(k) = zeta(12)*zeta(13) * Product_{p prime} (1 - 2/p^13 + 1/p^25) = 1.0001233729754... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^13). - Ridouane Oudra, Apr 02 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A161025 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 15.

Original entry on oeis.org

1, 16383, 2391484, 134209536, 1525878906, 39179682372, 113037178808, 1099444518912, 3812797945332, 24998474116998, 37974983358324, 320959957991424, 328114698808274, 1851888100411464, 3649114989636504
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2009

Keywords

Comments

a(n) is the number of lattices L in Z^14 such that the quotient group Z^14 / L is C_n. - Álvar Ibeas, Nov 26 2015

Links

Crossrefs

Column 14 of A263950.
Cf. A000010, A000203, A013671, A013672.

Programs

  • Maple
    A161025 := proc(n)
    add(numtheory[mobius](n/d)*d^14,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A161025(n)) ;
end do: # R. J. Mathar, Mar 15 2016
  • Mathematica
    A161025[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(15-1)/EulerPhi[n]&]
f[p_, e_] := p^(13*e - 13) * (p^14-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
  • PARI
    vector(100, n, sumdiv(n^13, d, if(ispower(d, 14), moebius(sqrtnint(d, 14))*sigma(n^13/d), 0))) \\ Altug Alkan, Nov 26 2015
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^14 - 1)*f[i,1]^(13*f[i,2] - 13)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

Formula

a(n) = J_14(n)/J_1(n) where J_14 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(13e-13) * (p^14-1) / (p-1).
For squarefree n, a(n) = A000203(n^13). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^14, where c = (1/14) * Product_{p prime} (1 + (p^13-1)/((p-1)*p^14)) = 0.1388226555... .
Sum_{k>=1} 1/a(k) = zeta(13)*zeta(14) * Product_{p prime} (1 - 2/p^14 + 1/p^27) = 1.00006146517418... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^14). - Ridouane Oudra, Apr 02 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A161139 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 16.

Original entry on oeis.org

1, 32767, 7174453, 536854528, 7629394531, 235085301451, 791260251657, 8795824586752, 34315186290957, 249992370597277, 417724816941565, 3851637578973184, 4265491084507563, 25927224666044919, 54736732481116543
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2009

Keywords

Comments

a(n) is the number of lattices L in Z^15 such that the quotient group Z^15 / L is C_n. - Álvar Ibeas, Nov 26 2015

Links

Crossrefs

Column 15 of A263950.
Cf. A000010, A000203. A013672, A013673.

Programs

  • Maple
    A161139 := proc(n)
    add(numtheory[mobius](n/d)*d^15,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A161139(n)) ;
end do: # R. J. Mathar, Mar 15 2016
  • Mathematica
    A161139[n_] := DivisorSum[n, MoebiusMu[n/#]*#^(16 - 1)/EulerPhi[n] &]; Array[A161139,20] (* Enrique Pérez Herrero, Mar 02 2011 *)
f[p_, e_] := p^(14*e - 14) * (p^15-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
  • PARI
    vector(100, n, sumdiv(n^14, d, if(ispower(d, 15), moebius(sqrtnint(d, 15))*sigma(n^14/d), 0))) \\ Altug Alkan, Nov 26 2015
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^15 - 1)*f[i,1]^(14*f[i,2] - 14)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

Formula

a(n) = J_15(n)/J_1(n), where J_15 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(14e-14) * (p^15-1) / (p-1).
For squarefree n, a(n) = A000203(n^14). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^15, where c = (1/15) * Product_{p prime} (1 + (p^14-1)/((p-1)*p^15)) = 0.1295704557... .
Sum_{k>=1} 1/a(k) = zeta(14)*zeta(15) * Product_{p prime} (1 - 2/p^15 + 1/p^29) = 1.00003065989236... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^15). - Ridouane Oudra, Apr 02 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A161167 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.

Original entry on oeis.org

1, 65535, 21523360, 2147450880, 38146972656, 1410533397600, 5538821761600, 70367670435840, 308836690967520, 2499961853010960, 4594972986357216, 46220358372556800, 55451384098598320, 362986684146456000, 821051025385244160, 2305807824841605120, 3041324492229179280
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2009

Keywords

Comments

a(n) is the number of lattices L in Z^16 such that the quotient group Z^16 / L is C_n. - Álvar Ibeas, Nov 26 2015

Links

Crossrefs

Column 16 of A263950.
Cf. A000010, A000203, A008683, A013673, A013674.

Programs

  • Maple
    A161167 := proc(n)
    add(numtheory[mobius](n/d)*d^16,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A161167(n)) ;
end do: # R. J. Mathar, Mar 15 2016
  • Mathematica
    A161167[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(17-1)/EulerPhi[n]&]; Array[A161167,20]
f[p_, e_] := p^(15*e - 15) * (p^16-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
  • PARI
    vector(100, n, sumdiv(n^15, d, if(ispower(d, 16), moebius(sqrtnint(d, 16))*sigma(n^15/d), 0))) \\ Altug Alkan, Nov 26 2015
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^16 - 1)*f[i,1]^(15*f[i,2] - 15)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

Formula

a(n) = J_16(n)/J_1(n) = J_16(n)/A000010(n), where J_k is the k-th Jordan totient function.
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(15e-15) * (p^16-1) / (p-1).
For squarefree n, a(n) = A000203(n^15). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^16, where c = (1/16) * Product_{p prime} (1 + (p^15-1)/((p-1)*p^16)) = 0.1214735403... .
Sum_{k>=1} 1/a(k) = zeta(15)*zeta(16) * Product_{p prime} (1 - 2/p^16 + 1/p^31) = 1.00001530597583... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^16). - Ridouane Oudra, Apr 02 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A344210 a(n) = Sum_{d|n} mu(n/d) * d^n / phi(n).

Original entry on oeis.org

1, 3, 13, 120, 781, 22932, 137257, 4177920, 64566801, 2497558338, 28531167061, 2228476723200, 25239592216021, 1851888100411464, 54736732481116543, 2305807824841605120, 51702516367896047761, 6557709646516945221396, 109912203092239643840221
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Crossrefs

Main diagonal of A263950.
Cf. A000010, A067858, A008683, A000203.

Programs

  • Mathematica
    Table[DivisorSum[n,MoebiusMu[n/#]*#^n/EulerPhi[n]&],{n,20}] (* Giorgos Kalogeropoulos, May 13 2021 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d^n)/eulerphi(n);

Formula

a(n) = J_n(n) / phi(n) = A067858(n) / A000010(n).
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^n). - Ridouane Oudra, Apr 03 2025
Previous Showing 11-15 of 15 results.

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

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