A330595 -id:A330595 - cp's OEIS frontend

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

7, 4, 8, 5, 3, 5, 2, 5, 9, 6, 8, 2, 3, 6, 3, 5, 6, 4, 6, 4, 4, 2, 1, 5, 0, 4, 8, 6, 3, 7, 9, 1, 0, 6, 0, 1, 6, 4, 1, 6, 4, 0, 3, 4, 3, 0, 0, 5, 3, 2, 4, 4, 0, 4, 5, 1, 5, 8, 5, 2, 7, 9, 3, 9, 2, 5, 9, 2, 5, 5, 8, 6, 8, 9, 5, 4, 9, 5, 8, 8, 3, 4, 2, 1, 2, 6, 2, 0, 6, 8, 1, 4, 6, 4, 7, 0, 9, 8, 1, 3, 1, 4, 3, 3, 5, 4

Offset: 0

Vaclav Kotesovec, Dec 19 2019

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 - 1/p^2 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

Original entry on oeis.org

1, 4, 8, 9, 25, 27, 36, 49, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 289, 343, 361, 392, 441, 484, 500, 529, 675, 676, 841, 900, 961, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444, 1521, 1681, 1764, 1800, 1849, 2116, 2197, 2209

Offset: 1

Amiram Eldar, Oct 22 2020

nonn

Equivalently, numbers k such that if a prime p divides k then p^2 divides k but p^4 does not divide k.
Each term has a unique representation as a^2 * b^3, where a and b are coprime squarefree numbers.
Dehkordi (1998) refers to these numbers as "2-full and 4-free numbers".

			4 = 2^2 is a term since the exponent of its only prime factor is 2.
72 = 2^3 * 3^2 is a terms since the exponents of the primes in its prime factorization are 2 and 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.
Eric Weisstein's World of Mathematics, Biquadratefree.
Eric Weisstein's World of Mathematics, Powerful Number.

Intersection of A001694 and A046100.
Subsequences: A062503, A062838.
Cf. A005117, A090699, A244000, A330595.

Select[Range[2500], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], MemberQ[{2, 3}, #1] &] &]

The number of terms not exceeding x is asymptotically (zeta(3/2)/zeta(3)) * J_2(1/2) * x^(1/2) + (zeta(2/3)/zeta(2)) * J_2(1/3) * x^(1/3), where J_2(s) = Product_{p prime} (1 - p^(-4*s) - p^(-5*s) - p^(-6*s) + p^(-7*s) + p^(-8*s)) (Dehkordi, 1998).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.748932... (A330595).

A064969 Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

Original entry on oeis.org

1, 8, 14, 36, 32, 112, 58, 148, 131, 256, 134, 504, 184, 464, 448, 596, 308, 1048, 382, 1152, 812, 1072, 554, 2072, 807, 1472, 1184, 2088, 872, 3584, 994, 2388, 1876, 2464, 1856, 4716, 1408, 3056, 2576, 4736, 1724, 6496, 1894, 4824, 4192, 4432, 2258, 8344

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 30 2001

Inverse Moebius transform of A160889. - Vladeta Jovovic, Nov 21 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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, 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.

Cf. A000010, A002117, A064803, A060648, A060724, A064767, A160889, A280184, A330595, A344219, A344302, A344303.

with(numtheory):
# define Jordan totient function J(r,n)
J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
seq(add(J(3,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := Sum[EulerPhi[i] EulerPhi[j] (EulerPhi[k] / EulerPhi[LCM[i, j, k]]), {i, Divisors[n]}, {j, Divisors[n]}, {k, Divisors[n]}];
Array[a, 48] (* Jean-François Alcover, Dec 13 2018, after Vladeta Jovovic *)
f[p_, e_] := 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, eulerphi(i)*eulerphi(j)*eulerphi(k)/eulerphi(lcm(lcm(i, j), k))))); \\ Michel Marcus, Dec 14 2018

a160889(n) = sumdiv(n, d, moebius(n/d)*d^3)/eulerphi(n);
a(n) = sumdiv(n, d, a160889(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{i|n, j|n, k|n} phi(i)*phi(j)*phi(k)/phi(lcm(i, j, k)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = A002117 * A330595 / 3 = 0.700772... . (End)
a(n) = Sum_{d divides n} J_3(d)/phi(d) = Sum_{1 <= i, j, k <= n} 1/phi(n/gcd(i,j,k,n)), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 23 2024

Formula and more terms from Vladeta Jovovic, Oct 30 2001

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

N. J. A. Sloane, Nov 19 2009

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

			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).

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.

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

Cf. A156304, A000203.

A160889[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(4-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Aug 22 2010 *)

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

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

Definition corrected by Vladeta Jovovic, Nov 21 2009

Typo in Mathematica program and formula fixed by Enrique Pérez Herrero, Oct 19 2010

A156304 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^3)*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 8, 21, 77, 199, 661, 1663, 4852, 12382, 33289, 82877, 213026, 518109, 1279852, 3053404, 7312985, 17093793, 39952528, 91661695, 209709116, 473095589, 1062567288, 2359804486, 5214774263, 11415904502, 24860918943, 53709881911

Offset: 0

Paul D. Hanna, Feb 08 2009

nonn

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

Euler transform of A160889. - Vaclav Kotesovec, Nov 01 2024

			G.f.: A(x) = 1 + x + 8*x^2 + 21*x^3 + 77*x^4 + 199*x^5 + 661*x^6 +...
log(A(x)) = x + 15*x^2/2 + 40*x^3/3 + 127*x^4/4 + 156*x^5/5 + 600*x^6/6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000203 (sigma), A000041 (partitions), A156303, A202993, A203557.

Cf. A160889, A175926, A330595.

{a(n)=polcoeff(exp(sum(m=1,n,sigma(m^3)*x^m/m)+x*O(x^n)),n)}

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

a(n) = (1/n)*Sum_{k=1..n} sigma(k^3) * a(n-k) for n>0, with a(0)=1.

log(a(n)) ~ 4*Pi*c^(1/4)*n^(3/4) / (3^(5/4)*5^(1/4)), where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.74893299784324530303390699... - Vaclav Kotesovec, Nov 01 2024

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

Original entry on oeis.org

1, 1, 0, 6, 9, 6, 0, 1, 1, 1, 9, 5, 3, 2, 1, 7, 6, 7, 6, 6, 5, 1, 1, 7, 9, 1, 3, 0, 0, 0, 7, 4, 3, 9, 5, 9, 2, 9, 4, 9, 5, 4, 8, 8, 3, 3, 6, 5, 8, 1, 2, 2, 4, 1, 9, 0, 4, 3, 1, 3, 4, 0, 4, 4, 9, 7, 8, 7, 7, 7, 3, 3, 2, 4, 1, 2, 3, 7, 3, 7, 0, 7, 8, 0, 4, 4, 4, 9, 8, 5, 6, 5, 9, 5, 9, 1, 2, 5, 3, 7, 2, 4, 9, 1, 6

Offset: 1

Vaclav Kotesovec, Dec 19 2019

			1.106960111953217676651179130007439592949548833658122419043134044978777...

Cf. A065464, A065473, A065476, A208133, A328017, A330595, A330596.

Do[Print[N[Exp[-Sum[q = Expand[(-p^2 + 2*p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 + 1/p^2 - 2/p^3) \\ Amiram Eldar, Mar 16 2021

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 1800, 2312, 2700, 2888, 3087, 3267, 3528, 4232, 4500, 4563, 5292, 5324, 5400, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10584, 10952, 11979, 12168, 12348, 13068, 13448, 13500, 14283, 14792

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {2, 3}.

Number k such that A051904(k) = 2 and A051903(k) = 3.

Equals A338325 \ (A062503 UNION A062838).
Subsequence of A001694 and A046100.
A143610 is a subsequence.
Cf. A051903, A051904, A136568.
Cf. A082020, A157289, A330595.

Select[Range[15000], Union[FactorInteger[#][[;; , 2]]] == {2, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [2, 3];
```

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) - 15/Pi^2 - zeta(3)/zeta(6) + 1 = A330595 - A082020 - A157289 + 1 = 0.047550294197921818806... .

A225959 a(n) = sigma(2*n^3) - sigma(n^3).

Original entry on oeis.org

2, 16, 80, 128, 312, 640, 800, 1024, 2186, 2496, 2928, 5120, 4760, 6400, 12480, 8192, 10440, 17488, 14480, 19968, 32000, 23424, 25440, 40960, 39062, 38080, 59048, 51200, 50520, 99840, 61568, 65536, 117120, 83520, 124800, 139904, 104120, 115840, 190400, 159744, 141288, 256000

Offset: 1

Paul D. Hanna, May 22 2013

nonn

Here sigma(n) = A000203(n), the sum of the divisors of n.

			L.g.f.: L(x) = 2*x + 16*x^2/2 + 80*x^3/3 + 128*x^4/4 + 312*x^5/5 + 640*x^6/6 +...
where
exp(L(x)) = 1 + 2*x + 10*x^2 + 44*x^3 + 134*x^4 + 468*x^5 + 1524*x^6 + 4584*x^7 + 13862*x^8 +...+ A225958(n)*x^n +...
exp(-L(-x)) = 1 + 2*x - 6*x^2 + 12*x^3 + 38*x^4 - 108*x^5 + 148*x^6 + 168*x^7 +...+ A225957(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A225958, A225957, A000203, A224903.

Cf. A013662, A330595.

a[n_] := DivisorSigma[1, 2*n^3] - DivisorSigma[1, n^3]; Array[a, 50] (* Amiram Eldar, Mar 17 2024 *)

{a(n)=sigma(2*n^3)-sigma(n^3)}
for(n=1, 50, print1(a(n), ", "))

a(n) = A054785(n^3).
Logarithmic derivative of A225958.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (15/44) * zeta(4) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = (15/44) * A013662 * A330595 = 0.64531050605789193162... . - Amiram Eldar, Mar 17 2024

A383292 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2s) + 1/p^(3s)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 4

Offset: 1

Vaclav Kotesovec, Apr 22 2025

First differs from A095691, A365552 and A368105 at n = 32.

The number of divisors of n that are both biquadratefree (A046100) and powerful (A001694). - Amiram Eldar, Apr 22 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A073184, A365498, A380922.
Cf. A095691, A365552, A368105.
Cf. A330595.
Cf. A046100, A001694.

f[p_, e_] := If[e < 4, e, 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 22 2025 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X^2 + X^3))[n], ", "))

Sum_{k=1..n} a(k) ~ c * n, where c = A330595 = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.74893299784324530303390699768511480225988349359548...

Multiplicative with a(p^e) = e if e < 4 and 3 otherwise. - Amiram Eldar, Apr 22 2025

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

1, 8, 1, 0, 7, 8, 1, 4, 7, 6, 1, 2, 1, 5, 6, 2, 9, 5, 2, 2, 4, 3, 1, 2, 5, 9, 0, 4, 4, 8, 6, 2, 5, 1, 8, 0, 8, 9, 7, 2, 5, 0, 3, 6, 1, 7, 9, 4, 5, 0, 0, 7, 2, 3, 5, 8, 9, 0, 0, 1, 4, 4, 7, 1, 7, 8, 0, 0, 2, 8, 9, 4, 3, 5, 6, 0, 0, 5, 7, 8, 8, 7, 1, 2, 0, 1, 1, 5, 7, 7, 4, 2, 4, 0, 2, 3, 1, 5, 4, 8, 4, 8, 0, 4, 6

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/(zeta(2*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

			1.8107814761215629522431259...

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Mathematica
```
RealDigits[N[5005/2764,105]][[1]]
```

default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

cp's OEIS Frontend

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

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A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

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A064969 Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

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A160889 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.

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A156304 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^3)*x^n/n ), a power series in x with integer coefficients.

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A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

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A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

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A383292 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2s) + 1/p^(3s)).