A013668 -id:A013668 - cp's OEIS frontend

A373103 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^4.

1, 497, 19603, 254449, 1952501, 9742691, 40351207, 130277873, 385845769, 970392997, 2357933051, 4987963747, 10604470813, 20054549879, 38274877103, 66702270961, 118587792977, 191765347193, 322687567459, 496811926949, 791004710821, 1171892726347, 1801152381623

Offset: 1

Seiichi Manyama, May 25 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A371491, A371878, A373007, A373105.
Cf. A350156, A372962, A372964.
Cf. A013664, A013668.

f[p_, e_] :=  (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 25 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 9));

a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^8*sigma(d^2, 8)/sigma(d^2, 4));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_9(d).
a(n) = Sum_{d|n} phi(n/d) * (n/d)^8 * sigma_8(d^2)/sigma_4(d^2).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1).
Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = zeta(10)/zeta(6) = Pi^4/99 = 0.983930212464... . (End)

A017681 Numerator of sum of -9th powers of divisors of n.

Original entry on oeis.org

1, 513, 19684, 262657, 1953126, 93499, 40353608, 134480385, 387440173, 500976819, 2357947692, 1292535097, 10604499374, 2587675113, 1423901192, 68853957121, 118587876498, 7361363287, 322687697780, 256501107891, 113474345696, 302406791499, 1801152661464, 24510295355

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 513/512, 19684/19683, 262657/262144, 1953126/1953125, 93499/93312, 40353608/40353607, 134480385/134217728, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017682 (denominator), A013667, A013668.

[Numerator(DivisorSigma(9,n)/n^9): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Table[Numerator[Total[1/Divisors[n]^9]],{n,20}] (* Harvey P. Dale, Aug 26 2013 *)
Table[Numerator[DivisorSigma[9, n]/n^9], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, numerator(sigma(n, 9)/n^9)) \\ G. C. Greubel, Nov 07 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^9*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017682(n) = zeta(9) (A013667).
Dirichlet g.f. of a(n)/A017682(n): zeta(s)*zeta(s+9).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017682(k) = zeta(10) (A013668). (End)

A017683 Numerator of sum of -10th powers of divisors of n.

Original entry on oeis.org

1, 1025, 59050, 1049601, 9765626, 30263125, 282475250, 1074791425, 3486843451, 200195333, 25937424602, 10329823175, 137858491850, 144768565625, 23066408612, 1100586419201, 2015993900450, 3574014537275, 6131066257802, 5125005407613, 16680163512500, 13292930108525

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 1025/1024, 59050/59049, 1049601/1048576, 9765626/9765625, 30263125/30233088, 282475250/282475249, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017684 (denominator), A013668, A013669.

[Numerator(DivisorSigma(10,n)/n^10): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Table[Numerator[Total[Divisors[n]^-10]],{n,20}] (* Harvey P. Dale, Sep 04 2018 *)
Table[Numerator[DivisorSigma[10, n]/n^10], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, numerator(sigma(n, 10)/n^10)) \\ G. C. Greubel, Nov 07 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^10*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017684(n) = zeta(10) (A013668).
Dirichlet g.f. of a(n)/A017684(n): zeta(s)*zeta(s+10).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017684(k) = zeta(11) (A013669). (End)

A157296 Decimal expansion of 31185/(2*Pi^8).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 26 2009

Equals the asymptotic mean of the abundancy index of the 9-free numbers (numbers that are not divisible by a 9th power other than 1) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			1.64329968185709999227... = (1+1/2^2+1/2^4+1/2^6+1/2^8)*(1+1/3^2+1/3^4+1/3^6+1/3^8)*(1+1/5^2+1/5^4+1/5^6+1/5^8)*...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Index entries for transcendental numbers.

Cf. A001248, A030514, A030514, A030516, A082020, A013661, A013668, A092748.

Maple
```
evalf(31185/2/Pi^8) ;
```

RealDigits[31185/(2*Pi^8),10,120][[1]] (* Harvey P. Dale, Mar 30 2018 *)

31185/2/Pi^8 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{p = primes = A000040} (1+1/p^2+1/p^4+1/p^6+1/p^8). The variant Product_{p} (1+1/p^2+1/p^6+1/p^8) equals A082020*Product_{p} (1+1/p^6) = A082020*zeta(6)/zeta(12) = 10135125/(691*Pi^8).

Equals A013661/A013668 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)+1/A030516(i)+1/A030514(i)^2) = 15592.5*A092748.

A161004 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 12.

Original entry on oeis.org

4095, 8382465, 362706435, 8583644160, 49987791945, 742460072445, 1349525501415, 8789651619840, 21417452280315, 102325010111415, 116835129114795, 760279114183680, 611574734464785, 2762478701396505, 4427568695944485, 9000603258716160, 8771463461234565

Offset: 1

N. J. A. Sloane, Nov 19 2009

nonn

Amiram Eldar, 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.

Cf. A000010, A013668, A013669, A160960.

f[p_, e_] := p^(10*e - 10) * (p^11-1) / (p-1); a[1] = 4095; a[n_] := 4095 * Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

a(n) = {my(f = factor(n)); 4095 * 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

From Amiram Eldar, Nov 08 2022: (Start)
a(n) = 4095 * A160960(n).
Sum_{k=1..n} a(k) ~ c * n^11, where c = (4095/11) * Product_{p prime} (1 + (p^10-1)/((p-1)*p^11)) =  723.3106628... .
Sum_{k>=1} 1/a(k) = (zeta(10)*zeta(11)/4095) * Product_{p prime} (1 - 2/p^11 + 1/p^21) = 0.0002443224366... . (End)

A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

Original entry on oeis.org

72, 108, 128, 200, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000

Offset: 1

V. Raman, Sep 07 2012

nonn

Terms of A216427 that are not 5th powers of squarefree numbers (A113850) and not 10th powers of primes (A030629). - Amiram Eldar, Feb 07 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A143610.
Subsequence of A216427. - Zak Seidov, Jan 03 2014
Cf. A002117, A013661, A013663, A013664, A013668, A030629, A085966, A113850.

With[{upto=4000},Select[Union[Flatten[{#[[1]]^2 #[[2]]^3,#[[2]]^2 #[[1]]^3}& /@ Subsets[Range[2,Surd[upto,2]],{2}]]],#<=upto&]](* Harvey P. Dale, Jan 04 2014 *)
pMx = 25; mx = 2^3 pMx^2; t = Flatten[Table[If[a != b, a^2 b^3, 0], {a, 2, mx^(1/2)}, {b, 2, mx^(1/3)}]]; Union[Select[t, 0 < # <= mx &]] (* T. D. Noe, Jan 02 2014 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4,3), for(a=2, sqrtint(lim\b^3), if(a!=b, listput(v, a^2*b^3)))); Set(v) \\ Charles R Greathouse IV, Jan 02 2014

from math import isqrt
from sympy import integer_nthroot, mobius, primepi
def A216426(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j, b, a, d = isqrt(x), integer_nthroot(x,6)[0], integer_nthroot(x,5)[0], integer_nthroot(x,10)[0]
        l, c = 0, n+x-2+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(d+1, b+1))+primepi(d)+sum(mobius(k)*(a//k**2+j//k**3) for k in range(1, d+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = sum(mobius(k)*((k2-1)//k**2) for k in range(1, isqrt(k2-1)+1))
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Sum_{n>=1} 1/a(n) = 2 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - zeta(5)/zeta(10) - P(6) - P(10) = 0.09117811499514578262..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

Name corrected by Charles R Greathouse IV, Jan 02 2014

A013686 Continued fraction for zeta(10).

Original entry on oeis.org

1, 1005, 2, 4, 1, 98, 7, 11, 2, 1, 1, 6, 2, 3, 28, 1, 37, 1, 2, 7, 9, 13, 85, 4, 3, 34, 5, 3, 7, 4, 7, 1, 3, 2, 1, 22, 1, 1, 1, 1, 3, 15, 1, 9, 12, 1, 3, 3, 3, 1, 3, 2, 1, 2, 1, 1, 2, 10, 8, 2, 2, 11, 54, 4, 5, 1, 2, 2, 1, 3, 2, 1, 19, 4, 5, 1, 2, 2, 7, 1, 200

Offset: 0

N. J. A. Sloane

Cf. A013668 (decimal expansion).

Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

ContinuedFraction[Zeta[10], 100] (* Paolo Xausa, Jul 03 2024 *)

Offset changed by Andrew Howroyd, Jul 09 2024

A160964 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 11.

Original entry on oeis.org

2047, 2094081, 60435628, 1072169472, 4997558082, 61825647444, 96371138776, 548950769664, 1189554465924, 5112501917886, 5309390815620, 31654731491328, 23516361067738, 98587674967848, 147547904812968, 281062794067968, 257921219638566, 1216914218640252

Offset: 1

N. J. A. Sloane, Nov 19 2009

nonn

Amiram Eldar, 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.

Cf. A000010, A013667, A013668, A160957.

f[p_, e_] := p^(9*e - 9) * (p^10-1) / (p-1); a[1] = 2047; a[n_] := 2047 * Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

a(n) = {my(f = factor(n)); 2047 * 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

From Amiram Eldar, Nov 08 2022: (Start)
a(n) = 2047 * A160957(n).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (2047/10) * Product_{p prime} (1 + (p^9-1)/((p-1)*p^10)) = 397.5922753... .
Sum_{k>=1} 1/a(k) = (zeta(9)*zeta(10)/2047) * Product_{p prime} (1 - 2/p^10 + 1/p^19) = 0.0004890150305... . (End)

A258331 Sum of the cubes of the divisors of n^3.

Original entry on oeis.org

1, 585, 20440, 299593, 1968876, 11957400, 40471600, 153391689, 402321277, 1151792460, 2359720584, 6123680920, 10609328380, 23675886000, 40243825440, 78536544841, 118612018980, 235357947045, 322734750520, 589861467468, 827239504000, 1380436541640

Offset: 1

Wesley Ivan Hurt, May 26 2015

			For n=2, the divisors of 2^3 = 8 are 1, 2, 4 and 8. The sum of the cubes of these divisors is 1^3+2^3+4^3+8^3 = 585, therefore a(2) = 585.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000578, A001158, A013668, A065827.

[DivisorSigma(3, n^3): n in [1..50]]; // Vincenzo Librandi, May 27 2015

with(numtheory): A258331:=n->sigma[3](n^3): seq(A258331(n), n=1..50);

Mathematica
```
Table[DivisorSigma[3, n^3], {n, 50}]
```

a(n)=sigma(n^3,3) \\ Charles R Greathouse IV, May 27 2015

from math import prod
from sympy import factorint
def A258331(n): return prod((p**((3*e+1)*3)-1)//(p**3-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

[sigma(n^3, 3) for n in (1..50)] # Bruno Berselli, May 27 2015

a(n) = sigma_3(n^3) = A001158(A000578(n)).
From Amiram Eldar, Nov 05 2022: (Start)
Multiplicative with a(p^e) = (p^(9*e + 3) - 1)/(p^3 - 1).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)/10) * Product_{p prime} (1 + 1/p^4 + 1/p^7) = 0.1087440273... . (End)

A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 2052, 177156, 4202512, 48828150, 363524112, 1977326792, 8606744640, 31382654013, 100195363800, 285311670732, 744500215872, 1792160394206, 4057474577184, 8650199741400, 17626613022976, 34271896307922, 64397206034676, 116490258898580, 205200886312800

Offset: 0

Seiichi Manyama, Feb 22 2017

Multiplicative because A013957 is. - Andrew Howroyd, Jul 23 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), A282753 (phi_{9, 2}), this sequence (phi_{11, 2}).
Cf. A282549 (E_2*E_4^3), A282792 (E_2^2*E_4*E_6), A282576 (E_2*E_6^2), A058550 (E_4^2*E_6 = E_14).
Cf. A013957 (sigma_9(n)), A282254 (n*sigma_9(n)), this sequence (n^2*sigma_9(n)).
Cf. A013668 (zeta(10)).

Table[If[n>0, n^2 * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

for(n=0, 20, print1(if(n==0, 0, n^2 * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

a(n) = n^2*A013957(n) for n > 0.
a(n) = (6*A282549(n) - 5*A282792(n) + 4*A282576(n) - 5*A058550(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(10) * n^12 / 12. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(9*e+9)-1)/(p^9-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-11). (End)

cp's OEIS Frontend

A373103 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^4.

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A017681 Numerator of sum of -9th powers of divisors of n.

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Magma

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A017683 Numerator of sum of -10th powers of divisors of n.

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Magma

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A157296 Decimal expansion of 31185/(2*Pi^8).

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A161004 a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 12.

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A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

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A013686 Continued fraction for zeta(10).

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A161004 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 12.

A160964 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 11.