A156302 -id:A156302 - cp's OEIS frontend

A060648 Number of cyclic subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).

1, 4, 5, 10, 7, 20, 9, 22, 17, 28, 13, 50, 15, 36, 35, 46, 19, 68, 21, 70, 45, 52, 25, 110, 37, 60, 53, 90, 31, 140, 33, 94, 65, 76, 63, 170, 39, 84, 75, 154, 43, 180, 45, 130, 119, 100, 49, 230, 65, 148, 95, 150, 55, 212, 91, 198, 105, 124, 61, 350, 63, 132, 153, 190

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001

The group U(n) of units modulo n acts on the direct product (Z_n)^k by multiplication. The number g(n,k) of orbits of U(n) acting on Z/(n)^k is g(n,k) = (1/phi(n))*Sum(gcd(n,a-1)^k) where the sum is over a in U(n) and phi(n) is the Euler totient function. A060648 gives g(n,2). - W. Edwin Clark, Jul 20 2001

a(n) is also the number of orbits of length n for the map TxT (Cartesion product) where T is a map with one orbit of each length. - Thomas Ward, Apr 08 2009

			The cycle index of C_4 X C_4 is (x(1)^4 + x(2)^2 + 2*x(4))^2 = x(1)^8 + 2*x(1)^4*x(2)^2 + 4*x(1)^4*x(4) + x(2)^4 + 4*x(2)^2*x(4) + 4*x(4)^2 and C_4 X C_4 has 1 element of order 1, 3 elements of order 2 and 12 elements of order 4. So a(4) = 1/phi(1) + 3/phi(2) + 12/phi(4) = 10, where phi = Euler totient function, cf. A000010. - _Vladeta Jovovic_, Jul 05 2001
For a(4) the pairs (e,d) are (1,4),(2,4),(4,4),(4,2),(4,1) with gcds 1,2,4,2,1 resp. giving 10 in total. - _Thomas Ward_, Apr 08 2009

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1211.1797 [math.GR], 2012-2014. - From _N. J. A. Sloane_, Jan 02 2013
W. G. Nowak and L. Tóth, On the average number of subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1307.1414 [math.NT], 2013.
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4.
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110, and arXiv:1103.5861, [math.NT], 2011.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013-2014.
Index entries for sequences related to groups.

Cf. A060724, A063379, A061503, A064969, A216620, A280184, A344219, A344302, A344303.

for n from 1 to 200 do:ans := 1:for i from 1 to nops(ifactors(n)[2]) do p := ifactors(n)[2][i][1]:e := ifactors(n)[2][i][2]:ans := ans*(p^(e+1)+p^e-2)/(p-1):od:printf(`%d,`,ans):od:

Table[ Plus @@ Map[ Times @@ (EulerPhi /@ #)/EulerPhi[ LCM @@ # ] &, Flatten[ Outer[ {##} &, Divisors[ i ], Divisors[ i ] ], 1 ] ], {i, 1, 100} ]
f[p_, e_] := (p^(e+1)+p^e-2)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

a(n) = sumdiv(n, d,  2^omega(d)*(n/d) ); \\ Joerg Arndt, Sep 16 2012

def A060648(n) :
    def dedekind_psi(n) : return n*mul(1+1/p for p in prime_divisors(n))
    return reduce(lambda x,y: x+y, [dedekind_psi(d) for d in divisors(n)])
[A060648(n) for n in (1..64)]  # Peter Luschny, Sep 10 2012

a(n) is multiplicative: if the canonical factorization of n is the product of p^e(p) over primes then a(n) = product a(p^e(p)). If n = p^e, p prime, a(n) = (p^(e+1)+p^e-2)/(p-1).
a(n) = Sum_{i|n, j|n} phi(i)*phi(j)/phi(lcm(i, j)). - Vladeta Jovovic, Jul 07 2001
a(n) = Sum_{i|n, j|n} phi(gcd(i, j)).
a(n) = Sum_{d|n} phi(n/d)*tau(d^2).
a(n) = sum(d|n, sigma(d)*moebius(n/d)^2 ). - Benoit Cloitre, Sep 08 2002
Inverse Euler transform of A156302. - Vladeta Jovovic, Feb 14 2009
Moebius transform of A060724. - Vladeta Jovovic, Apr 05 2009
Also a(n) = (1/n)*Sum_{d|n} sigma(d)^2*moebius(n/d). - Vladeta Jovovic, Mar 31 2009
Inverse Moebius transform of A001615. - Vladeta Jovovic, Apr 05 2009
From Thomas Ward, Apr 08 2009: (Start)
a(n) = Sum_{lcm(e,d)=n} gcd(e,d).
Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2s). (End)
For the proofs of these formulas see the papers of L. Toth.
a(n) = Sum_{d|n} psi(d), where psi is Dedekind's psi function A001615. - Peter Luschny, Sep 10 2012
a(n) = Sum_{d|n} 2^omega(d)*(n/d). - Peter Luschny, Sep 15 2012
Sum_{k=1..n} a(k) ~ (5/4) * n^2. - Amiram Eldar, Oct 19 2022
a(n) = Sum_{k=1..n} tau(gcd(n,k)^2). - Ridouane Oudra, Apr 10 2023
a(n) = Sum_{d divides n} J_2(d)/phi(d) = Sum_{1 <= i, j <= n} 1/phi(n/gcd(i,j,n)), where the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 23 2024

More terms and additional comments from Vladeta Jovovic, Jul 05 2001

A178933 Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).

Original entry on oeis.org

1, 1, 14, 35, 205, 521, 2507, 6709, 26712, 73834, 262431, 724537, 2384988, 6552033, 20289864, 55244988, 163342701, 439201501, 1251532060, 3321188863, 9177476977, 24028568664, 64709650590, 167153761523, 440300702427, 1122562426240, 2900254892900, 7301575351055, 18544013542057

Offset: 0

Joerg Arndt, Dec 30 2010

nonn

Compare with g.f. for partition numbers A000041: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Similarly, exp( Sum_{n>=1} sigma(n)^2*x^n/n ) gives A156302.

			G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 521*x^5 + 2507*x^6 +...
such that, by definition,
log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 6^3*x^5/5 + 12^3*x^6/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..6700

Cf. A000203 (sigma), A000041 (partitions), A156302, A205797, A361147.

nmax = 30; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1,k]^3 * a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 30 2024 *)

N=100;v=Vec(exp(sum(k=1,N,sigma(k)^3*x^k/k)+x*O(x^N)))

a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^3*a(n-k)))

{a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^3*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */

{a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^2*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */

a(0)=0 and a(n)=1/n*sum(k=1,n,sigma(k)^3*a(n-k)) for n>0.
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^2 * x^(n*k) / n ). [Paul D. Hanna, Jan 31 2012]
From Vaclav Kotesovec, Oct 30 2024: (Start)
log(a(n)) ~ 2^(7/4) * c^(1/4) * Pi^(3/2) * zeta(3)^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)), where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.8359835743341928677044245715803848964089818378791769798895797934086403174189...
Equivalently, log(a(n)) ~ 3.2753680082113515869730831738879060384726246... * n^(3/4). (End)

A205797 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^4 * x^n/n ).

Original entry on oeis.org

1, 1, 41, 126, 1526, 5185, 46920, 176865, 1254608, 4986548, 30563031, 123868761, 683127011, 2793828323, 14223836013, 58127497582, 278433541834, 1130954381904, 5159127957638, 20767403083249, 91032595281699, 362455763000997, 1536849042738162

Offset: 0

Paul D. Hanna, Jan 31 2012

nonn

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

			G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 5185*x^5 +...
such that, by definition,
log(A(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 6^4*x^5/5 + 12^4*x^6/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..3200

Cf. A156302, A178933, A000203 (sigma), A000041 (partitions), A361179.

nmax = 30; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^4 * a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 30 2024 *)

{a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^4*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */

{a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^3*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */

a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^4*a(n-k)))

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^4*a(n-k) for n>0, with a(0) = 1.
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^3 * x^(n*k) / n ).
From Vaclav Kotesovec, Oct 30 2024: (Start)
log(a(n)) ~ 5^(4/5) * c^(1/5) * Pi^(6/5) * zeta(3)^(1/5) * zeta(5)^(1/5) * n^(4/5) / (2^(9/5) * 3^(2/5)), where c = Product_{primes p} (1 + 3/p^2 + 5/p^3 + 3/p^4 + 3/p^5 + 5/p^6 + 3/p^7 + 1/p^9) = 6.04468280906514379869287397833397910321972833863778...
Equivalently, log(a(n)) ~ 3.967005157823944635858584839447899089435134... * n^(4/5). (End)

A382125 G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n) * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

1, 3, 15, 52, 180, 555, 1696, 4809, 13410, 35844, 93771, 238305, 594403, 1449441, 3476607, 8190824, 19015548, 43492230, 98197506, 218885763, 482337864, 1051051262, 2266904481, 4840955055, 10242621395, 21479302368, 44666897613, 92139573135, 188617118541, 383280793962, 773395096907

Offset: 0

Paul D. Hanna, Apr 06 2025

nonn

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Equals the self-convolution cube of A382124.
Conjectures: a(3*n) == A382124(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0.

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + 35844*x^9 + 93771*x^10 + ...
where
A(x) = exp(3*x + 21*x^2/2 + 48*x^3/3 + 105*x^4/4 + 108*x^5/5 + 336*x^6/6 + 192*x^7/7 + 465*x^8/8 + 507*x^9/9 + 756*x^10/10 + ... + sigma(n)*sigma(2*n)*x^n/n + ...).
RELATED SERIES.
A(x)^(1/3) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + 44*x^5 + 105*x^6 + 200*x^7 + 425*x^8 + 825*x^9 + 1634*x^10 + ... + A382124(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A382124, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).

Cf. A087943, A329963.

nmax=30; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/n ,{n,nmax}]],{x,0,nmax}],x] (* Stefano Spezia, Apr 06 2025 *)

{a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)*x^m/m ) +x*O(x^n) ));
polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
(2) A(x) = exp( Sum_{n>=1} Sum_{k>=1} sigma(2*n*k) * x^(n*k) / n ).
(3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k) * a(n-k) for n>0, with a(0) = 1.

A163658 G.f.: A(x) = exp( Sum_{n>=1} A163659(n)^2x^n/n ), where xexp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 1, 5, 6, 26, 30, 95, 115, 347, 412, 1076, 1308, 3277, 3941, 9081, 11050, 24694, 29834, 63067, 76711, 158127, 191360, 379032, 460448, 893441, 1081113, 2035189, 2468182, 4565994, 5520070, 9970503, 12068315, 21475803, 25926236, 45246532

Offset: 0

Paul D. Hanna, Aug 02 2009

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 6*x^3 + 26*x^4 + 30*x^5 + 95*x^6 +...
log(A(X)) = x + 3^2*x^2/2 + 2^2*x^3/3 + 7^2*x^4/4 + x^5/5 + 6^2*x^6/6 +...
log(S(x)/x) = x + 3*x^2/2 - 2*x^3/3 + 7*x^4/4 + x^5/5 - 6*x^6/6 +...
where S(x) is the g.f. of Stern's diatomic series (A002487):
S(x) = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + 4*x^9 +...

Cf. A163659, A002487, A156302 (variant).

{A002487(n)=local(c=1, b=0); while(n>0, if(bitand(n, 1), b+=c, c+=b); n>>=1); b}
{A163659(n)=n*polcoeff(log(sum(k=0,n,A002487(k+1)*x^k)+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A163659(k)^2*x^k/k)+x*O(x^n)), n)}

A377507 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^2 * x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 12, 66, 690, 4860, 63000, 711900, 8876700, 131405400, 2160219600, 37553808600, 686750664600, 13805424032400, 278759396916000, 6445702905642000, 150985820419434000, 3825993309462324000, 99427990563910008000, 2724045313186016820000, 78032929885709378580000

Offset: 0

Vaclav Kotesovec, Oct 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..435

Cf. A318917, A377508, A377509.

Cf. A065464, A127473, A156302.

nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^2*a[n-k], {k, 1, n}]/n];Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^2 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

log(a(n)/n!) ~ 3 * c^(1/3) * n^(2/3) / 2^(2/3), where c = Product_{p primes} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298...

A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

1, 1, 4, 9, 22, 44, 105, 200, 425, 825, 1634, 3072, 5917, 10846, 20153, 36436, 65882, 116831, 207293, 361502, 629539, 1083068, 1856251, 3150554, 5328137, 8933266, 14920357, 24745481, 40869317, 67089425, 109697089, 178379353, 288953043, 465805681, 748079686, 1196148976, 1905801579, 3024212984

Offset: 0

Paul D. Hanna, Apr 06 2025

nonn

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Equals the self-convolution cube root of A382125.
Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0.

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + 44*x^5 + 105*x^6 + 200*x^7 + 425*x^8 + 825*x^9 + 1634*x^10 + 3072*x^11 + 5917*x^12 + ...
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + ... + A382125(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A382125, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).

Cf. A087943, A329963.

nmax=37; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/(3n) ,{n,nmax}]],{x,0,nmax}],x] (* Stefano Spezia, Apr 06 2025 *)

{a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)/3*x^m/m ) +x*O(x^n) ));
polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( (1/3) * Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
(2) A(x) = exp( (1/3) * Sum_{n>=1} (1/n) * Sum_{k>=1} sigma(2*n*k) * x^(n*k) ).
(3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k)/3 * a(n-k) for n > 0, with a(0) = 1.

A195734 G.f.: exp( Sum_{n>=1} (2sigma(n^2) - sigma(n)^2) x^n/n ).

Original entry on oeis.org

1, 1, 3, 6, 11, 22, 40, 72, 123, 215, 363, 605, 991, 1618, 2598, 4139, 6537, 10229, 15871, 24476, 37487, 56995, 86177, 129531, 193662, 287992, 426254, 627841, 920708, 1344331, 1954987, 2831688, 4086168, 5875087, 8417724, 12020250, 17108958, 24275947, 34340966

Offset: 0

Paul D. Hanna, Sep 22 2011

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 22*x^5 + 40*x^6 + 72*x^7 +...
where
log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 13*x^4/4 + 26*x^5/5 + 38*x^6/6 + 50*x^7/7 + 29*x^8/8 +...+ A195735(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)

Cf. A195735 (log), A156302, A156303, A000203.

nmax = 40; CoefficientList[Series[Exp[Sum[(2*DivisorSigma[1,k^2] - DivisorSigma[1,k]^2) * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 31 2024 *)

{a(n)=polcoeff(exp(sum(k=1, n,(2*sigma(k^2)-sigma(k)^2)*x^k/k)+x*O(x^n)), n)}

Logarithmic derivative equals A195735.

log(a(n)) ~ 3*(5*zeta(3)*(12 - Pi^2))^(1/3) * n^(2/3) / (2*Pi^(2/3)). - Vaclav Kotesovec, Oct 31 2024

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

Paul D. Hanna, Oct 10 2010

nonn

sigma(n) = A000203(n), sum of divisors of n;

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

			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 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

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

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 *)

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(m)^2+sigma(m,2))/2*x^m/m)+x*O(x^n)), n)}

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

cp's OEIS Frontend

A060648 Number of cyclic subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).

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A178933 Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).

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A205797 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^4 * x^n/n ).

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A382125 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

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A163658 G.f.: A(x) = exp( Sum_{n>=1} A163659(n)^2*x^n/n ), where x*exp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

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A377507 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^2 * x^k/k), where phi is the Euler totient function A000010.

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A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

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A382125 G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n) * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

A163658 G.f.: A(x) = exp( Sum_{n>=1} A163659(n)^2x^n/n ), where xexp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

A195734 G.f.: exp( Sum_{n>=1} (2sigma(n^2) - sigma(n)^2) x^n/n ).