A013674 -id:A013674 - cp's OEIS frontend

A013963 a(n) = sigma_15(n), the sum of the 15th powers of the divisors of n.

1, 32769, 14348908, 1073774593, 30517578126, 470199366252, 4747561509944, 35185445863425, 205891146443557, 1000030517610894, 4177248169415652, 15407492847694444, 51185893014090758, 155572843119354936, 437893920912786408, 1152956690052710401, 2862423051509815794

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013674, A013954-A013972, A017665-A017712.

[DivisorSigma(15, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[15, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^15*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 15); \\ Amiram Eldar, Oct 29 2023

[sigma(n,15)for n in range(1,15)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^15*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-15)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(15*e+15)-1)/(p^15-1).
Sum_{k=1..n} a(k) = zeta(16) * n^16 / 16 + O(n^17). (End)

A293904 Decimal expansion of zeta(21).

Original entry on oeis.org

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

Offset: 1

Frank Ellermann, Oct 19 2017

Web searches find 1.0000004769329867878 in Python tools. Simon Plouffe published 1000 digits for zeta(9) up to zeta(2051) many years ago.

			1.000000476932986787806...

Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each.
Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each. [Cached copy, with permission]
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each. (gzipped file)
Simon Plouffe, Identities inspired from Ramanujan Notebooks II, July 21, 1998.

Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678.

RealDigits[Zeta[21], 10, 100][[1]] (* Amiram Eldar, May 31 2021 *)

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

A010804 16th powers: a(n) = n^16.

Original entry on oeis.org

0, 1, 65536, 43046721, 4294967296, 152587890625, 2821109907456, 33232930569601, 281474976710656, 1853020188851841, 10000000000000000, 45949729863572161, 184884258895036416, 665416609183179841, 2177953337809371136, 6568408355712890625, 18446744073709551616, 48661191875666868481

Offset: 0

N. J. A. Sloane

Exponent towers of the form n^(2^(2^2)). - Paul Duckett, Aug 30 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A013674 (zeta(16)).

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A001016 (8th powers), A008456 (12th powers).

[n^16: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

A010804 := n -> n^16; # M. F. Hasler, Jul 03 2025

Range[0, 15]^16 (* Alonso del Arte, Feb 16 2015 *)

A010804(n):=n^16$
makelist(A010804(n),n,0,10); /* Martin Ettl, Nov 12 2012 */

A010804(n)=n^16 \\ Charles R Greathouse IV, Jun 28 2015

A010804 = lambda n: n**16 # M. F. Hasler, Jul 03 2025

Completely multiplicative with a(p) = p^16 for prime p. Multiplicative with a(p^e) = p^(16e). - Jaroslav Krizek, Nov 01 2009
From Ilya Gutkovskiy, Feb 27 2017: (Start)
Dirichlet g.f.: zeta(s-16).
Sum_{n>=1} 1/a(n) =  3617*Pi^16/325641566250 = A013674. (End)
a(n) = A001016(n)^2. - Michel Marcus, Feb 28 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = 32767*zeta(16)/32768 = 16931177*Pi^16/1524374691840000. - Amiram Eldar, Oct 08 2020

A056555 Smallest number k (k>0) such that n*k is a perfect 4th power.

Original entry on oeis.org

1, 8, 27, 4, 125, 216, 343, 2, 9, 1000, 1331, 108, 2197, 2744, 3375, 1, 4913, 72, 6859, 500, 9261, 10648, 12167, 54, 25, 17576, 3, 1372, 24389, 27000, 29791, 8, 35937, 39304, 42875, 36, 50653, 54872, 59319, 250, 68921, 74088, 79507, 5324, 1125

Offset: 1

Henry Bottomley, Jun 25 2000

			a(64) = 4 because the smallest 4th power divisible by 64 is 256 and 64*4 = 256.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A000190, A008835, A048798, A053164, A053165, A053166, A053167, A056554.

Cf. A013662, A013674.

f[p_, e_] := p^Mod[4 - e, 4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 08 2020 *)

a(n,f=factor(n))=f[,2]=-f[,2]%4; factorback(f) \\ Charles R Greathouse IV, Apr 24 2020

a(n) = A053167(n)/n = n^3/A000190(n)^4 = A056553(n)/A053165(n).
Multiplicative with a(p^e) = p^((4 - e) mod 4). - Amiram Eldar, Sep 08 2020
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(16)/(4*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.1537848996... . - Amiram Eldar, Oct 27 2022

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

N. J. A. Sloane, Nov 19 2009

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

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 16 of A263950.

Cf. A000010, A000203, A008683, A013673, A013674.

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

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 *)

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

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

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

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A017693 Numerator of sum of -15th powers of divisors of n.

Original entry on oeis.org

1, 32769, 14348908, 1073774593, 30517578126, 13061093507, 4747561509944, 35185445863425, 205891146443557, 500015258805447, 4177248169415652, 3851873211923611, 51185893014090758, 19446605389919367, 48654880101420712, 1152956690052710401, 2862423051509815794

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

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

Cf. A017694 (denominator), A013673, A013674.

[Numerator(DivisorSigma(15,n)/n^15): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Numerator[DivisorSigma[15, n]/n^15], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 15)/n^15)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017694(n) = zeta(15) (A013673).
Dirichlet g.f. of a(n)/A017694(n): zeta(s)*zeta(s+15).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017694(k) = zeta(16) (A013674). (End)

A017695 Numerator of sum of -16th powers of divisors of n.

Original entry on oeis.org

1, 65537, 43046722, 4295032833, 152587890626, 1410576509857, 33232930569602, 281479271743489, 1853020231898563, 5000076293978081, 45949729863572162, 30814514057170571, 665416609183179842, 1088993285370003137, 6568408508343827972, 18447025552981295105, 48661191875666868482

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

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

Cf. A017696 (denominator), A013674, A013675.

[Numerator(DivisorSigma(16,n)/n^16): n in [1..20]]; // G. C. Greubel, Nov 05 2018

Table[Numerator[Total[1/Divisors[n]^16]],{n,20}] (* Harvey P. Dale, Sep 26 2014 *)
Table[Numerator[DivisorSigma[16, n]/n^16], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)

vector(20, n, numerator(sigma(n, 16)/n^16)) \\ G. C. Greubel, Nov 05 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017696(n) = zeta(16) (A013674).
Dirichlet g.f. of a(n)/A017696(n): zeta(s)*zeta(s+16).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017696(k) = zeta(17) (A013675). (End)

A161215 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 17.

Original entry on oeis.org

131071, 8589737985, 2821088318560, 281468534292480, 4999961852994576, 184880022956829600, 725978907114673600, 9223160931695984640, 40479533921803813920, 327672500035999538160, 602267704294826658336, 6058148592249392332800, 7268068365187380400720

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, A013673, A013674, A161167.

f[p_, e_] := p^(15*e - 15) * (p^16-1) / (p-1); a[1] = 131071; a[n_] := 131071 * Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

a(n) = {my(f = factor(n)); 131071 * 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

From Amiram Eldar, Nov 08 2022: (Start)
a(n) = 131071 * A161167(n).
Sum_{k=1..n} a(k) ~ c * n^16, where c = (131071/16) * Product_{p prime} (1 + (p^15-1)/((p-1)*p^16)) = 15921.65841... .
Sum_{k>=1} 1/a(k) = (zeta(15)*zeta(16)/131071) * Product_{p prime} (1 - 2/p^16 + 1/p^31) = 7.6295695155...*10^(-6). (End)

A282777 Expansion of phi_{16, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 65538, 43046724, 4295098372, 152587890630, 2821196197512, 33232930569608, 281483566907400, 1853020317992013, 10000305176108940, 45949729863572172, 184889914172333328, 665416609183179854, 2178019803670969104, 6568408813691796120

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A013963 is. - Andrew Howroyd, Jul 25 2018

George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012. See p. 212.

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), A282548 (phi_{12, 1}), A282597 (phi_{14, 1}), this sequence (phi_{16, 1}).
Cf. A282546 (E_2*E_4^4), A282000 (E_4^3*E_6), A282547 (E_2*E_4*E_6^2), A282253 (E_6^3).
Cf. A013674.

Table[If[n==0, 0, n * DivisorSigma[15, n]], {n, 0, 15}] (* Indranil Ghosh, Mar 11 2017 *)

for(n=0, 15, print1(if(n==0, 0, n * sigma(n, 15)), ", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = n*A013963(n) for n > 0.
a(n) = (2156*A282546(n) - 4156*A282000(n) + 8000*A282547(n)/3 - 2000*A282253(n)/3)/16320.
Sum_{k=1..n} a(k) ~ zeta(16) * n^17 / 17. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(15*e+15)-1)/(p^15-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-16). (End)

cp's OEIS Frontend

A013963 a(n) = sigma_15(n), the sum of the 15th powers of the divisors of n.

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A293904 Decimal expansion of zeta(21).

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

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A010804 16th powers: a(n) = n^16.

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A056555 Smallest number k (k>0) such that n*k is a perfect 4th power.

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

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A161215 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 17.