A013669 -id:A013669 - cp's OEIS frontend

A017685 Numerator of sum of -11th powers of divisors of n.

1, 2049, 177148, 4196353, 48828126, 30248021, 1977326744, 8594130945, 31381236757, 50024415087, 285311670612, 185843885311, 1792160394038, 506442812307, 2883268288216, 17600780175361, 34271896307634, 21433384705031, 116490258898220, 102450026512239, 350279478046112

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. A017686 (denominator), A013669, A013670.

[Numerator(DivisorSigma(11,n)/n^11): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Numerator[Total[Divisors[n]^-11]],{n,20}] (* Harvey P. Dale, Aug 26 2012 *)
Table[Numerator[DivisorSigma[11, n]/n^11], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 11)/n^11)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017686(n) = zeta(11) (A013669).
Dirichlet g.f. of a(n)/A017686(n): zeta(s)*zeta(s+11).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017686(k) = zeta(12) (A013670). (End)

A321819 a(n) = Sum_{d|n, n/d odd} d^10 for n > 0.

Original entry on oeis.org

1, 1024, 59050, 1048576, 9765626, 60467200, 282475250, 1073741824, 3486843451, 10000001024, 25937424602, 61918412800, 137858491850, 289254656000, 576660215300, 1099511627776, 2015993900450, 3570527693824, 6131066257802, 10240001048576

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A321543 - A321565, A321807 - A321836 for related sequences.

Cf. A013669.

a[n_] := DivisorSum[n, #^10 &, OddQ[n/#] &]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)

apply( A321819(n)=sumdiv(n,d,if(bittest(n\d,0),d^10)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^10*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(10*e) and a(p^e) = (p^(10*e+10)-1)/(p^10-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^11, where c = 2047*zeta(11)/22528 = 0.090909606... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-10)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A016823 a(n) = (4n+1)^11.

Original entry on oeis.org

1, 48828125, 31381059609, 1792160394037, 34271896307633, 350277500542221, 2384185791015625, 12200509765705829, 50542106513726817, 177917621779460413, 550329031716248441, 1532278301220703125, 3909821048582988049, 9269035929372191597, 20635899893042801193

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A001020, A013669, A016813.

[(4*n+1)^11 : n in [0..20]]; // Wesley Ivan Hurt, Oct 10 2014

A016823:=n->(4*n+1)^11: seq(A016823(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014

Table[(4 n + 1)^11, {n, 0, 20}] (* Wesley Ivan Hurt, Oct 10 2014 *)
CoefficientList[Series[(1 + 48828113 x + 30795122175 x^2 + 1418810334759 x^3 + 14826379326378 x^4 + 50417667664170 x^5 + 64020606756990 x^6 + 31088834650350 x^7 + 5356480404741 x^8 + 261595441397 x^9 + 1975200979 x^10 + 177147 x^11)/(x - 1)^12, {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 10 2014 *)

From Wesley Ivan Hurt, Oct 10 2014 : (Start)
G.f.: (1 + 48828113*x + 30795122175*x^2 + 1418810334759*x^3 + 14826379326378*x^4 + 50417667664170*x^5 + 64020606756990*x^6 + 31088834650350*x^7 + 5356480404741*x^8 + 261595441397*x^9 + 1975200979*x^10 + 177147*x^11) / (x - 1)^12.
Recurrence: a(n) = 12*a(n-1)-66*a(n-2)+220*a(n-3)-495*a(n-4)+792*a(n-5)-924*a(n-6)+792*a(n-7)-495*a(n-8)+220*a(n-9)-66*a(n-10)+12*a(n-11)-a(n-12).
a(n) = A016813(n)^11 = A001020(A016813(n)). (End)
Sum_{n>=0} 1/a(n) = 50521*Pi^11/29727129600 + 2047*zeta(11)/4096. - Amiram Eldar, Apr 21 2023

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)

A309929 Decimal expansion of Pi^11/Zeta(11).

Original entry on oeis.org

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

Offset: 6

Seiichi Manyama, Aug 23 2019

Index entries for zeta function.

Cf. A013669, A308637.

RealDigits[Pi^11/Zeta[11], 10, 100][[1]] (* Amiram Eldar, Aug 24 2019 *)

PARI
```
Pi^11/zeta(11)
```

More terms from Amiram Eldar, Aug 24 2019

A016835 a(n) = (4n+2)^11.

Original entry on oeis.org

2048, 362797056, 100000000000, 4049565169664, 64268410079232, 584318301411328, 3670344486987776, 17714700000000000, 70188843638032384, 238572050223552512, 717368321110468608, 1951354384207722496, 4882812500000000000, 11384956040305711104, 24986644000165537792

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A013669, A016763, A016825.

Table[(4*n+2)^11, {n, 0, 20}] (* Amiram Eldar, Apr 21 2023 *)

From Amiram Eldar, Apr 21 2023: (Start)
a(n) = A016825(n)^11.
a(n) = 2^11*A016763(n).
Sum_{n>=0} 1/a(n) = 2047*zeta(11)/4194304.
Sum_{n>=0} (-1)^n/a(n) = 50521*Pi^11/30440580710400. (End)

A017099 a(n) = (8*n + 2)^11.

Original entry on oeis.org

2048, 100000000000, 64268410079232, 3670344486987776, 70188843638032384, 717368321110468608, 4882812500000000000, 24986644000165537792, 103510234140112521216, 364375289404334925824, 1127073856954876807168, 3138105960900000000000, 8007313507497959524352

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A013669, A016823, A017089.

[(8*n+2)^11: n in [0..20]]; // Vincenzo Librandi, Jul 12 2011

(8*Range[0,20]+2)^11 (* Harvey P. Dale, Dec 12 2012 *)

From Amiram Eldar, Apr 24 2023: (Start)
a(n) = A017089(n)^11.
a(n) = 2^11*A016823(n).
Sum_{n>=0} 1/a(n) = 50521*Pi^11/60881161420800 + 2047*zeta(11)/8388608. (End)
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12). - Wesley Ivan Hurt, Jan 20 2024

A017123 a(n) = (8*n + 4)^11.

Original entry on oeis.org

4194304, 743008370688, 204800000000000, 8293509467471872, 131621703842267136, 1196683881290399744, 7516865509350965248, 36279705600000000000, 143746751770690322432, 488595558857835544576, 1469170321634239709184, 3996373778857415671808, 10000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A013669, A016763, A016835, A017113.

[(8*n+4)^11: n in [0..15] ]; // Vincenzo Librandi, Jul 21 2011

(8*Range[0,20]+4)^11 (* Harvey P. Dale, Jun 11 2016 *)

G.f.: ( 4194304*(1+x)*(x^10 + 177134*x^9 + 46525293*x^8 + 1356555432*x^7 + 9480267666*x^6 + 19107752148*x^5 + 9480267666*x^4 + 1356555432*x^3 + 46525293*x^2 + 177134*x+1) ) / ( (x-1)^12 ). - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^11.
a(n) = 2^11*A016835(n) = 2^22*A016763(n).
Sum_{n>=0} 1/a(n) = 2047*zeta(11)/8589934592.
Sum_{n>=0} (-1)^n/a(n) = 50521*Pi^11/62342309294899200. (End)

A017147 a(n) = (8*n+6)^11.

Original entry on oeis.org

362797056, 4049565169664, 584318301411328, 17714700000000000, 238572050223552512, 1951354384207722496, 11384956040305711104, 52036560683837093888, 197732674300000000000, 650190514836423555072, 1903193578437064103936, 5062982072492057196544, 12433743083946522728448

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A013669, A016847, A017137.

[(8*n+6)^11: n in [0..15]]; // Vincenzo Librandi, Jul 22 2011

(8*Range[0,20]+6)^11 (* Harvey P. Dale, Apr 30 2018 *)

a(n) = 2048*A016847(n). - R. J. Mathar, Aug 26 2015
From Amiram Eldar, Apr 26 2023: (Start)
a(n) = A017137(n)^11.
Sum_{n>=0} 1/a(n) = 2047*zeta(11)/8388608 - 50521*Pi^11/60881161420800. (End)

A017207 a(n) = (9*n + 3)^11.

Original entry on oeis.org

177147, 743008370688, 350277500542221, 17714700000000000, 317475837322472439, 3116402981210161152, 20635899893042801193, 103510234140112521216, 422351360321044921875, 1469170321634239709184, 4501035456767426597157, 12433743083946522728448, 31517572945366073781711

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A013669, A016787, A017197.

[(9*n+3)^11: n in [0..15]]; // Vincenzo Librandi, Jul 23 2011

Table[(9*n + 3)^11, {n, 0, 15}] (* Amiram Eldar, Oct 03 2024 *)

From Amiram Eldar, Oct 03 2024: (Start)
a(n) = A017197(n)^11 = 3^11 * A016787(n).
Sum_{n>=0} 1/a(n) = 7388*Pi^11/(444826519957575*sqrt(3)) + 88573*zeta(11)/31381059609.  (End)

cp's OEIS Frontend

A017685 Numerator of sum of -11th powers of divisors of n.

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A321819 a(n) = Sum_{d|n, n/d odd} d^10 for n > 0.

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A016823 a(n) = (4n+1)^11.

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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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A309929 Decimal expansion of Pi^11/Zeta(11).

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A016835 a(n) = (4n+2)^11.

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A017099 a(n) = (8*n + 2)^11.

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Magma

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A017123 a(n) = (8*n + 4)^11.

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