A013672 -id:A013672 - cp's OEIS frontend

A113852 Numbers whose prime factors are raised to the seventh power.

128, 2187, 78125, 279936, 823543, 10000000, 19487171, 62748517, 105413504, 170859375, 410338673, 893871739, 1801088541, 2494357888, 3404825447, 8031810176, 17249876309, 21870000000, 27512614111, 42618442977, 52523350144, 64339296875, 94931877133, 114415582592

Offset: 1

Cino Hilliard, Jan 25 2006

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

Proper subset of A001015.
Nonunit terms of A329332 column 7 in ascending order.
Cf. A005117, A013665, A013672.

Select[Range@34^7, Union[Last /@ FactorInteger@# ] == {7} &] (* Robert G. Wilson v, Jan 26 2006 *)
Select[Range[2, 34], SquareFreeQ]^7 (* Amiram Eldar, Oct 13 2020 *)

allpwrfact(n,p) = /* All prime factors are raised to the power p */ { local(x,j,ln,y,flag); for(x=4,n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1,ln, if(y[2][j]==p,flag++); ); if(flag==ln,print1(x",")); ) }

from math import isqrt
from sympy import mobius
def A113852(n):
    def f(x): return int(n+1-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**7 # Chai Wah Wu, Feb 25 2025

From Amiram Eldar, Oct 13 2020: (Start)
a(n) = A005117(n+1)^7.
Sum_{n>=1} 1/a(n) = zeta(7)/zeta(14) - 1. (End)

More terms from Robert G. Wilson v, Jan 26 2006

A010802 14th powers: a(n) = n^14.

Original entry on oeis.org

0, 1, 16384, 4782969, 268435456, 6103515625, 78364164096, 678223072849, 4398046511104, 22876792454961, 100000000000000, 379749833583241, 1283918464548864, 3937376385699289, 11112006825558016, 29192926025390625, 72057594037927936, 168377826559400929, 374813367582081024

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A013672 (zeta(14)), A001015 (n^7).

Cf. A000290, (squares), A000578, (cubes), A000583, (4th powers), A000584, (5th powers), A008455 (11th powers).

[n^14: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Range[0,20]^14 (* Harvey P. Dale, Nov 08 2011 *)

for(n=0,15,print1(n^14,", ")) \\ Derek Orr, Feb 27 2017

A010802(n)=n^14 \\ M. F. Hasler, Jul 03 2025

A010802 = lambda n: n**14 # M. F. Hasler, Jul 03 2025

Totally multiplicative with a(p) = p^14 for prime p. Multiplicative with a(p^e) = p^(14e). - Jaroslav Krizek, Nov 01 2009
From Ilya Gutkovskiy, Feb 27 2017: (Start)
Dirichlet g.f.: zeta(s-14).
Sum_{n>=1} 1/a(n) = 2*Pi^14/18243225 = A013672. (End)
a(n) = A001015(n)^2. - Michel Marcus, Feb 28 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = 8191*zeta(14)/8192 = 8191*Pi^14/74724249600. - Amiram Eldar, Oct 08 2020

A161025 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 15.

Original entry on oeis.org

1, 16383, 2391484, 134209536, 1525878906, 39179682372, 113037178808, 1099444518912, 3812797945332, 24998474116998, 37974983358324, 320959957991424, 328114698808274, 1851888100411464, 3649114989636504

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^14 such that the quotient group Z^14 / 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 14 of A263950.

Cf. A000010, A000203, A013671, A013672.

A161025 := proc(n)
    add(numtheory[mobius](n/d)*d^14,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A161025(n)) ;
end do: # R. J. Mathar, Mar 15 2016

A161025[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(15-1)/EulerPhi[n]&]
f[p_, e_] := p^(13*e - 13) * (p^14-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^13, d, if(ispower(d, 14), moebius(sqrtnint(d, 14))*sigma(n^13/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^14 - 1)*f[i,1]^(13*f[i,2] - 13)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_14(n)/J_1(n) where J_14 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(13e-13) * (p^14-1) / (p-1).
For squarefree n, a(n) = A000203(n^13). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^14, where c = (1/14) * Product_{p prime} (1 + (p^13-1)/((p-1)*p^14)) = 0.1388226555... .
Sum_{k>=1} 1/a(k) = zeta(13)*zeta(14) * Product_{p prime} (1 - 2/p^14 + 1/p^27) = 1.00006146517418... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^14). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A161139 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 16.

Original entry on oeis.org

1, 32767, 7174453, 536854528, 7629394531, 235085301451, 791260251657, 8795824586752, 34315186290957, 249992370597277, 417724816941565, 3851637578973184, 4265491084507563, 25927224666044919, 54736732481116543

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^15 such that the quotient group Z^15 / 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 15 of A263950.

Cf. A000010, A000203. A013672, A013673.

A161139 := proc(n)
    add(numtheory[mobius](n/d)*d^15,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A161139(n)) ;
end do: # R. J. Mathar, Mar 15 2016

A161139[n_] := DivisorSum[n, MoebiusMu[n/#]*#^(16 - 1)/EulerPhi[n] &]; Array[A161139,20] (* Enrique Pérez Herrero, Mar 02 2011 *)
f[p_, e_] := p^(14*e - 14) * (p^15-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^14, d, if(ispower(d, 15), moebius(sqrtnint(d, 15))*sigma(n^14/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^15 - 1)*f[i,1]^(14*f[i,2] - 14)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_15(n)/J_1(n), where J_15 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(14e-14) * (p^15-1) / (p-1).
For squarefree n, a(n) = A000203(n^14). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^15, where c = (1/15) * Product_{p prime} (1 + (p^14-1)/((p-1)*p^15)) = 0.1295704557... .
Sum_{k>=1} 1/a(k) = zeta(14)*zeta(15) * Product_{p prime} (1 - 2/p^15 + 1/p^29) = 1.00003065989236... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^15). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A017689 Numerator of sum of -13th powers of divisors of n.

Original entry on oeis.org

1, 8193, 1594324, 67117057, 1220703126, 1088524711, 96889010408, 549822930945, 2541867422653, 5000610355659, 34522712143932, 26751583696117, 302875106592254, 99226457784093, 648732096885608, 4504149450301441, 9904578032905938, 6941839931265343, 42052983462257060

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. A017690 (denominator), A013671, A013672.

[Numerator(DivisorSigma(13,n)/n^13): n in [1..20]]; // G. C. Greubel, Nov 06 2018

A017689 := proc(n)
    numtheory[sigma][-13](n) ;
    numer(%) ;
end proc: # R. J. Mathar, Sep 21 2017

Table[Numerator[DivisorSigma[13, n]/n^13], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 13)/n^13)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017690(n) = zeta(13) (A013671).
Dirichlet g.f. of a(n)/A017690(n): zeta(s)*zeta(s+13).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017690(k) = zeta(14) (A013672). (End)

A017691 Numerator of sum of -14th powers of divisors of n.

Original entry on oeis.org

1, 16385, 4782970, 268451841, 6103515626, 39184481725, 678223072850, 4398314962945, 22876797237931, 10000610353201, 379749833583242, 213999516991295, 3937376385699290, 5556342524323625, 5838586426737844, 72061992352890881, 168377826559400930, 374836322743499435

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. A017692 (denominator), A013672, A013673.

[Numerator(DivisorSigma(14,n)/n^14): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Table[Numerator[DivisorSigma[14, n]/n^14], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

vector(20, n, numerator(sigma(n, 14)/n^14)) \\ G. C. Greubel, Nov 06 2018

From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017692(n) = zeta(14) (A013672).
Dirichlet g.f. of a(n)/A017692(n): zeta(s)*zeta(s+14).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017692(k) = zeta(15) (A013673). (End)

A282597 Expansion of phi_{14, 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, 16386, 4782972, 268468228, 6103515630, 78373779192, 678223072856, 4398583447560, 22876806803877, 100012207113180, 379749833583252, 1284076017413616, 3937376385699302, 11113363271818416, 29192944359852360, 72066391204823056, 168377826559400946

Offset: 0

Seiichi Manyama, Feb 19 2017

Multiplicative because A013961 is. - Andrew Howroyd, Jul 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

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}), this sequence (phi_{14, 1}).
Cf. A282012 (E_4^4), A282287 (E_4*E_6^2), A282596 (E_2*E_4^2*E_6).
Cf. A013672.

Table[n * DivisorSigma[13, n], {n, 0, 17}] (* Amiram Eldar, Sep 06 2023 *)

a(n) = if(n < 1, 0, n*sigma(n, 13)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n*A013961(n) for n > 0.
a(n) = (3*A282012(n) + 4*A282287(n) - 7*A282596(n))/144.
Sum_{k=1..n} a(k) ~ zeta(14) * n^15 / 15. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(13*e+13)-1)/(p^13-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-14). (End)

A161195 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 16.

Original entry on oeis.org

65535, 2147385345, 470177777355, 35182761492480, 499992370589085, 15406315230591285, 51855240592341495, 576434364292792320, 2248845733577866995, 16383250007092548195, 27375595878265462275, 252417068738007613440, 279538958223203141205, 1699140668489253766665

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, A013672, A013673, A161139.

f[p_, e_] := p^(14*e - 14) * (p^15-1) / (p-1); a[1] = 65535; a[n_] := 65535* Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

a(n) = {my(f = factor(n)); 65535 * prod(i = 1, #f~, (f[i,1]^15 - 1)*f[i,1]^(14*f[i,2] - 14)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

From Amiram Eldar, Nov 08 2022: (Start)
a(n) = 65535 * A161139(n).
Sum_{k=1..n} a(k) ~ c * n^15, where c = 4369 * Product_{p prime} (1 + (p^14-1)/((p-1)*p^15)) = 8491.399817... .
Sum_{k>=1} 1/a(k) = (zeta(14)*zeta(15)/65535) * Product_{p prime} (1 - 2/p^15 + 1/p^29) = 1.5259489736...*10^(-5). (End)

A161157 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 15.

Original entry on oeis.org

32767, 536821761, 78361756228, 4397643866112, 49998474112902, 1283800652283324, 3703889238001736, 36025498551189504, 124933950274693644, 819125001391673466, 1244326279702202508, 10516894943504990208, 10751334335850714158, 60680817386182440888

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, A013671, A013672, A161025.

f[p_, e_] := p^(13*e - 13) * (p^14-1) / (p-1); a[1] = 32767; a[n_] := 32767 * Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

a(n) = {my(f = factor(n)); 32767 * prod(i = 1, #f~, (f[i,1]^14 - 1)*f[i,1]^(13*f[i,2] - 13)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

From Amiram Eldar, Nov 08 2022: (Start)
a(n) = 32767 * A161025(n).
Sum_{k=1..n} a(k) ~ c * n^14, where c = (4681/2) * Product_{p prime} (1 + (p^13-1)/((p-1)*p^14)) =  4548.801953... .
Sum_{k>=1} 1/a(k) = (zeta(13)*zeta(14)/32767) * Product_{p prime} (1 - 2/p^14 + 1/p^27) = 3.05203853014...*10^(-5). (End)

A013690 Continued fraction for zeta(14).

Original entry on oeis.org

1, 16327, 36, 19, 2, 1, 35, 1, 4, 7, 5, 1, 1, 1, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 2, 1, 3, 1, 1, 7, 1, 4, 7, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 4, 9, 2, 2, 1, 23, 6, 1, 2, 1, 2, 1, 1, 10, 1, 19, 7, 1, 1, 42, 1, 15, 1, 1, 4, 1, 2, 2, 1

Offset: 0

N. J. A. Sloane

Cf. A013672.

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

ContinuedFraction[Zeta[14],80] (* Harvey P. Dale, Jun 28 2014 *)

Offset changed by Andrew Howroyd, Jul 08 2024

cp's OEIS Frontend

A113852 Numbers whose prime factors are raised to the seventh power.

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A010802 14th powers: a(n) = n^14.

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A161025 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 15.

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A161139 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 16.

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A017689 Numerator of sum of -13th powers of divisors of n.

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Magma

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A017691 Numerator of sum of -14th powers of divisors of n.

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A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

A161195 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 16.

A161157 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 15.