A263950 -id:A263950 - cp's OEIS frontend

A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

1, 127, 1093, 8128, 19531, 138811, 137257, 520192, 796797, 2480437, 1948717, 8883904, 5229043, 17431639, 21347383, 33292288, 25646167, 101193219, 49659541, 158747968, 150021901, 247487059, 154764793, 568569856, 305171875, 664088461, 580865013, 1115624896

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^7 such that the quotient group Z^7 / L is C_n. - Álvar Ibeas, Oct 30 2015

Enrique Pérez Herrero and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..5000 from Enrique Pérez Herrero)
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 7 of A263950.

Cf. A000010, A000203, A008683, A013664, A013665, A069092.

A160897 := proc(n)
    add(numtheory[mobius](n/d)*d^7,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A160897(n)) ;
end do: # R. J. Mathar, Mar 14 2016

A160897[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(8 - 1)/EulerPhi[n] &] (* Enrique Pérez Herrero, Oct 27 2010 *)
f[p_, e_] := p^(6*e - 6) * (p^7-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(30, n, sumdiv(n^6, d, if(ispower(d, 7), moebius(sqrtnint(d, 7))*sigma(n^6/d), 0))) \\ Altug Alkan, Oct 30 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(6*f[i,2]-6)*(1+p+p^2+p^3+p^4+p^5+p^6); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_7(n)/J_1(n) = J_7(n)/phi(n) = A069092(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 27 2010
From Álvar Ibeas, Oct 30 2015: (Start)
Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).
For squarefree n, a(n) = A000203(n^6). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 + (p^6-1)/((p-1)*p^7)) = 0.2761554804... .
Sum_{k>=1} 1/a(k) = zeta(6)*zeta(7) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 1.008982290854... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^7). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 27 2010

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

N. J. A. Sloane, Nov 19 2009

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

			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.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 4 of A263950.

Cf. A000010, A000203, A059377, A160870.

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

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 *)

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

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

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

Definition corrected by Enrique Pérez Herrero, Aug 22 2010

A160893 a(n) = Sum_{d|n} Möbius(n/d)*d^5/phi(n).

Original entry on oeis.org

1, 31, 121, 496, 781, 3751, 2801, 7936, 9801, 24211, 16105, 60016, 30941, 86831, 94501, 126976, 88741, 303831, 137561, 387376, 338921, 499255, 292561, 960256, 488125, 959171, 793881, 1389296, 732541, 2929531, 954305, 2031616, 1948705

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^5 such that the quotient group Z^5 / L is C_nm x (C_m)^4 (and also (C_nm)^4 x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 5 of A263950.

Cf. A000010, A000203, A013662, A013663, A059378.

A160893 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(4*e-4)*(1+p+p^2+p^3+p^4) ; end do; a; end proc: # R. J. Mathar, Jul 10 2011

A160893[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(6-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 19 2010 *)
f[p_, e_] := p^(4*e - 4)*(1 + p + p^2 + p^3 + p^4); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 08 2022 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^5/eulerphi(n)); \\ Michel Marcus, Feb 15 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^5 - 1)*f[i, 1]^(4*f[i, 2] - 4)/(f[i, 1] - 1)); } \\ Amiram Eldar, Nov 08 2022

a(n) = J_5(n)/J_1(n) = J_5(n)/phi(n) = A059378(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^(4e-4)*(1 + p+ p^2 + p^3 + p^4). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^4). - Álvar Ibeas, Oct 30 2015
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^5, where c = (1/5) * Product_{p prime} (1 + (p^4-1)/((p-1)*p^5)) = 0.3799167034... .
Sum_{k>=1} 1/a(k) = zeta(4)*zeta(5) * Product_{p prime} (1 - 2/p^5 + 1/p^9) = 1.0449010968... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^5). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 19 2010

A160895 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 7.

Original entry on oeis.org

1, 63, 364, 2016, 3906, 22932, 19608, 64512, 88452, 246078, 177156, 733824, 402234, 1235304, 1421784, 2064384, 1508598, 5572476, 2613660, 7874496, 7137312, 11160828, 6728904, 23482368, 12206250, 25340742, 21493836, 39529728, 21243690, 89572392

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^6 such that the quotient group Z^6 / L is C_nm x (C_m)^5 (and also (C_nm)^5 x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 6 of A263950.

Cf. A000010, A000203, A013663, A013664, A069091.

A160895 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(5*e-5)*(1+p+p^2+p^3+p^4+p^5) ; end do; a; end proc: # R. J. Mathar, Jul 10 2011

A160895[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(7-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 20 2010 *)
f[p_, e_] := p^(5*e - 5) * (p^6-1) / (p-1); ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 08 2022 *)

vector(50, n, sumdiv(n^5, d, if(ispower(d, 6), moebius(sqrtnint(d, 6))*sigma(n^5/d), 0))) \\ Altug Alkan, Oct 30 2014

a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(5*f[i,2]-5)*(1+p+p^2+p^3+p^4+p^5); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_6(n)/J_1(n)=J_6(n)/phi(n)=A069091(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 20 2010
Multiplicative with a(p^e) = p^(5e-5)*(1+p+p^2+p^3+p^4+p^5). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^5). - Álvar Ibeas, Oct 30 2015
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^6, where c = (1/6) * Product_{p prime} (1 + (p^5-1)/((p-1)*p^6)) = 0.3203646372... .
Sum_{k>=1} 1/a(k) = zeta(5)*zeta(6) * Product_{p prime} (1 - 2/p^6 + 1/p^11) = 1.0195114923... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^6). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 20 2010

A161213 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 18.

Original entry on oeis.org

1, 131071, 64570081, 8589869056, 190734863281, 8463265086751, 38771752331201, 562945658454016, 2779530261754401, 24999809265103951, 50544702849929377, 554648540725313536, 720867993281778161, 5081852349802846271, 12315765571578095761, 36893206672442392576

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^17 such that the quotient group Z^17 / L is C_n. - Álvar Ibeas, Nov 26 2015

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 17 of A263950.

Cf. A000010, A000203, A008683, A013674, A013675.

A161213 := proc(n)
    add(numtheory[mobius](n/d)*d^17,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A161213(n)) ;
end do: # R. J. Mathar, Mar 15 2016

A161213[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(18-1)/EulerPhi[n]&]; Array[A161213,20]
f[p_, e_] := p^(16*e - 16) * (p^17-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

A161213(n)=sumdiv(n,d,moebius(n/d)*d^17)/eulerphi(n);

vector(100, n, sumdiv(n^16, d, if(ispower(d, 17), moebius(sqrtnint(d, 16))*sigma(n^16/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^17 - 1)*f[i,1]^(16*f[i,2] - 16)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_17(n)/A000010(n), where J_17 is the 17th Jordan totient function.
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(16e-16) * (p^17-1) / (p-1).
For squarefree n, a(n) = A000203(n^16). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} (1 + (p^16-1)/((p-1)*p^17)) = 0.1143286202... .
Sum_{k>=1} 1/a(k) = zeta(16)*zeta(17) * Product_{p prime} (1 - 2/p^17 + 1/p^33) = 1.000007645061593... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^17). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A160908 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.

Original entry on oeis.org

1, 255, 3280, 32640, 97656, 836400, 960800, 4177920, 7173360, 24902280, 21435888, 107059200, 67977560, 245004000, 320311680, 534773760, 435984840, 1829206800, 943531280, 3187491840, 3151424000, 5466151440, 3559590240, 13703577600, 7629375000, 17334277800

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^8 such that the quotient group Z^8 / L is C_n. - Álvar Ibeas, Oct 30 2015

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 8 of A263950.

Cf. A000010, A000203, A008683, A013665, A013666, A069093.

A160908[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(9-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 28 2010 *)
f[p_, e_] := p^(7*e - 7) * (p^8-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(30, n, sumdiv(n^7, d, if(ispower(d, 8), moebius(sqrtnint(d, 8))*sigma(n^7/d), 0))) \\ Altug Alkan, Oct 30 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; f[i,1] = p^(7*f[i,2]-7)*(p^8-1)/(p-1); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_8(n)/J_1(n) = J_8(n)/phi(n) = A069093(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 28 2010
From Álvar Ibeas, Oct 30 2015: (Start)
Multiplicative with a(p^e) = p^(7e-7) * (p^8-1) / (p-1).
For squarefree n, a(n) = A000203(n^7). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^8, where c = (1/8) * Product_{p prime} (1 + (p^7-1)/((p-1)*p^8)) = 0.2423008904... .
Sum_{k>=1} 1/a(k) = zeta(7)*zeta(8) * Product_{p prime} (1 - 2/p^8 + 1/p^15) = 1.004270064601... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^8). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 28 2010

A160953 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.

Original entry on oeis.org

1, 511, 9841, 130816, 488281, 5028751, 6725601, 33488896, 64566801, 249511591, 235794769, 1287360256, 883708281, 3436782111, 4805173321, 8573157376, 7411742281, 32993635311, 17927094321, 63874967296, 66186639441, 120491126959, 81870575521, 329564225536

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^9 such that the quotient group Z^9 / L is C_n. - Álvar Ibeas, Nov 03 2015

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Enrique Pérez Herrero, Mathematica Package: Jordan Totient Function.
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 9 of A263950.

Cf. A000010, A000203, A008683, A013666, A013667, A069094.

A160953 := proc(n)
    add(numtheory[mobius](n/d)*d^9,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A160953(n)) ;
end do: # R. J. Mathar, Mar 14 2016

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A160953[n_]:=JordanTotient[n,9]/JordanTotient[n];
f[p_, e_] := p^(8*e - 8) * (p^9-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^8, d, if(ispower(d, 9), moebius(sqrtnint(d, 9))*sigma(n^8/d), 0))) \\ Altug Alkan, Nov 05 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; f[i,1] = p^(8*f[i,2]-8)*(p^9-1)/(p-1); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_9(n)/phi(n) = A069094(n)/A000010(n).
From Álvar Ibeas, Nov 03 2015: (Start)
Multiplicative with a(p^e) = p^(8e-8) * (p^9-1) / (p-1).
For squarefree n, a(n) = A000203(n^8). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 + (p^8-1)/((p-1)*p^9)) = 0.2156692448... .
Sum_{k>=1} 1/a(k) = zeta(8)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^17) = 1.002068659133... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^9). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A160957 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 11.

Original entry on oeis.org

1, 1023, 29524, 523776, 2441406, 30203052, 47079208, 268173312, 581120892, 2497558338, 2593742460, 15463962624, 11488207654, 48162029784, 72080070744, 137304735744, 125999618778, 594486672516, 340614792100, 1278749869056

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^10 such that the quotient group Z^10 / L is C_n. - Álvar Ibeas, Nov 26 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 10 of A263950.

Cf. A000010, A000203, A013667, A013668, A069095.

b = 11; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 20}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(9*e - 9) * (p^10-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^9, d, if(ispower(d, 10), moebius(sqrtnint(d, 10))*sigma(n^9/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^10 - 1)*f[i,1]^(9*f[i,2] - 9)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = A069095(n)/A000010(n). - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(9e-9) * (p^10-1) / (p-1).
For squarefree n, a(n) = A000203(n^9). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^10, where c = (1/10) * Product_{p prime} (1 + (p^9-1)/((p-1)*p^10)) = 0.1942316928... .
Sum_{k>=1} 1/a(k) = zeta(9)*zeta(10) * Product_{p prime} (1 - 2/p^10 + 1/p^19) = 1.0010137674... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^10). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A160960 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 12.

Original entry on oeis.org

1, 2047, 88573, 2096128, 12207031, 181308931, 329554457, 2146435072, 5230147077, 24987792457, 28531167061, 185660345344, 149346699503, 674597973479, 1081213356763, 2197949513728, 2141993519227, 10706111066619, 6471681049901, 25587499475968, 29189626919861, 58403298973867

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^11 such that the quotient group Z^11 / L is C_n. - Álvar Ibeas, Nov 26 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 11 of A263950.

Cf. A000010, A000203, A008683, A013668, A013669.

b = 12; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 18}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(10*e - 10) * (p^11-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^10, d, if(ispower(d, 11), moebius(sqrtnint(d, 11))*sigma(n^10/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^11 - 1)*f[i,1]^(10*f[i,2] - 10)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_11(n)/J_1(n) where J_11 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^(10e-10) * (p^11-1) / (p-1).
For squarefree n, a(n) = A000203(n^10). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^11, where c = (1/11) * Product_{p prime} (1 + (p^10-1)/((p-1)*p^11)) = 0.1766326404... .
Sum_{k>=1} 1/a(k) = zeta(10)*zeta(11) * Product_{p prime} (1 - 2/p^11 + 1/p^21) = 1.0005003781952... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^11). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A160972 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 13.

Original entry on oeis.org

1, 4095, 265720, 8386560, 61035156, 1088123400, 2306881200, 17175674880, 47071500840, 249938963820, 313842837672, 2228476723200, 1941507093540, 9446678514000, 16218261652320, 35175782154240, 36413889826860, 192757795939800, 122961939948120, 511874997903360, 612984472464000

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^12 such that the quotient group Z^12 / L is C_n. - Álvar Ibeas, Nov 26 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000
Jin Ho Kwak and Jaeun 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.
Index to Jordan function ratios J_k/J_1.

Column 12 of A263950.

Cf. A000010, A000203, A008683, A013669, A013670.

b = 13; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 17}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(11*e - 11) * (p^12-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^11, d, if(ispower(d, 12), moebius(sqrtnint(d, 12))*sigma(n^11/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^12 - 1)*f[i,1]^(11*f[i,2] - 11)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_12(n)/J_1(n) where J_12 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^(11e-11) * (p^12-1) / (p-1).
For squarefree n, a(n) = A000203(n^11). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^12, where c = (1/12) * Product_{p prime} (1 + (p^11-1)/((p-1)*p^12)) = 0.1619398772... .
Sum_{k>=1} 1/a(k) = zeta(11)*zeta(12) * Product_{p prime} (1 - 2/p^12 + 1/p^23) = 1.0002481006668... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^12). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

cp's OEIS Frontend

A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A160893 a(n) = Sum_{d|n} Möbius(n/d)*d^5/phi(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A160895 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A161213 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 18.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A160908 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.

Views

Author

Keywords

Comments

Links

Crossrefs