A069092 -id:A069092 - cp's OEIS frontend

A000541 Sum of 7th powers: 1^7 + 2^7 + ... + n^7.

0, 1, 129, 2316, 18700, 96825, 376761, 1200304, 3297456, 8080425, 18080425, 37567596, 73399404, 136147921, 241561425, 412420800, 680856256, 1091194929, 1703414961, 2597286700, 3877286700, 5678375241, 8172733129, 11577558576, 16164030000, 22267545625

Offset: 0

N. J. A. Sloane

a(n) is divisible by A000537(n) if and only n is congruent to 1 mod 3 (see A016777) - Artur Jasinski, Oct 10 2007

This sequence is related to A000540 by a(n) = n*A000540(n) - Sum_{i=0..n-1} A000540(i). - Bruno Berselli, Apr 26 2010

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 815.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row 7 of array A103438.

Cf. A000217, A000537, A000539, A069092.

[n^2*(n+1)^2*(3*n^4+6*n^3-n^2-4*n+2)/24: n in [0..30]]; // Vincenzo Librandi, Feb 20 2016

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^7 od: seq(a[n], n=0..25); # Zerinvary Lajos, Feb 22 2008

Table[Sum[k^7, {k, 1, n}], {n, 0, 100}] (* Artur Jasinski, Oct 10 2007 *)
s = 0; lst = {s}; Do[s += n^7; AppendTo[lst, s], {n, 1, 30, 1}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 1, 129, 2316, 18700, 96825, 376761, 1200304, 3297456}, 35] (* Vincenzo Librandi, Feb 20 2016 *)

a(n)=n^2*(n+1)^2*(3*n^4+6*n^3-n^2-4*n+2)/24 \\ Edward Jiang, Sep 10 2014

a(n) = sum(i=1, n, i^7); \\ Michel Marcus, Sep 11 2014

A000541_list, m = [0], [5040, -15120, 16800, -8400, 1806, -126, 1, 0, 0]
for _ in range(10**2):
    for i in range(8):
        m[i+1] += m[i]
    A000541_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

a(n) = n^2*(n+1)^2*(3*n^4 + 6*n^3 - n^2 - 4*n + 2)/24.
a(n) = sqrt(Sum_{j=1..n} Sum_{i=1..n} (i*j)^7). - Alexander Adamchuk, Oct 26 2004
Jacobi formula: a(n) = 2(A000217(n))^4 - A000539(n). - Artur Jasinski, Oct 10 2007
G.f.: x*(1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1-x)^9. - Colin Barker, May 25 2012
a(n) = 8*a(n-1) - 28* a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 5040. - Ant King, Sep 24 2013
a(n) = -Sum_{j=1..7} j*Stirling1(n+1,n+1-j)*Stirling2(n+7-j,n). - Mircea Merca, Jan 25 2014
a(n) = 2*A000217(n)^4 - (4/3)*A000217(n)^3 + (1/3)*A000217(n)^2. - Michael Raney, Feb 19 2016
a(n) = 72*A288876(n-2) + 48*A006542(n+2) + A000537(n). - Yasser Arath Chavez Reyes, Apr 27 2024
a(n) = Sum_{i=1..n} J_7(i)*floor(n/i), where J_7 is A069092. - Ridouane Oudra, Jul 17 2025

A069091 Jordan function J_6(n).

Original entry on oeis.org

1, 63, 728, 4032, 15624, 45864, 117648, 258048, 530712, 984312, 1771560, 2935296, 4826808, 7411824, 11374272, 16515072, 24137568, 33434856, 47045880, 62995968, 85647744, 111608280, 148035888, 187858944, 244125000, 304088904, 386889048

Offset: 1

Benoit Cloitre, Apr 05 2002

Moebius transform of n^6. - Enrique Pérez Herrero, Sep 14 2010

a(n) is divisible by 504 = (2^3)*(3^3)*7 = A006863(3) except for n = 1, 2, 3 and 7. See Lugo. - Peter Bala, Jan 13 2024

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Michael Lugo, A little number theory problem (2008)
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5), A069092 - A069095 (J_7 through J_10).
Cf. A065959.
Cf. A013665.

with(numtheory): seq(add(d^6 * mobius(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 13 2024

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]
A069091[n_IntegerQ]:=JordanTotient[n,6]; (* Enrique Pérez Herrero, Sep 14 2010 *)
f[p_, e_] := p^(6*e) - p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(sumdiv(n,d,d^6*moebius(n/d)),","))

a(n) = Sum_{d|n} d^6*mu(n/d).
Multiplicative with a(p^e) = p^(6e)-p^(6(e-1)).
Dirichlet generating function: zeta(s-6)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^6*Product_{distinct primes p dividing n} (1-1/p^6). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^7 / (7*zeta(7)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^6 = 1/zeta(7).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^6/(p^6-1)^2) = 1.0175973008... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A(x^n)/(1 - x^n)^7 = x + 63*x^2 + 728*x^3 + 4032*x^4 + 15624*x^5 + ..., where A(x) = x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6 is the 6th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A351247 a(n) = n^7 * Sum_{p|n, p prime} 1/p^7.

Original entry on oeis.org

0, 1, 1, 128, 1, 2315, 1, 16384, 2187, 78253, 1, 296320, 1, 823671, 80312, 2097152, 1, 5062905, 1, 10016384, 825730, 19487299, 1, 37928960, 78125, 62748645, 4782969, 105429888, 1, 181139311, 1, 268435456, 19489358, 410338801, 901668, 648051840, 1, 893871867, 62750704

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^7. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 2315; a(6) = 6^7 * Sum_{p|6, p prime} 1/p^7 = 279936 * (1/2^7 + 1/3^7) = 2315.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), this sequence (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069092.

Array[#^7*DivisorSum[#, 1/#^7 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069092(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^7, where c = A010051.
a(p^k) = p^(7*k-7) for p prime and k>=1. (End)

A351302 a(n) = n^7 * Product_{p|n, p prime} (1 + 1/p^7).

Original entry on oeis.org

1, 129, 2188, 16512, 78126, 282252, 823544, 2113536, 4785156, 10078254, 19487172, 36128256, 62748518, 106237176, 170939688, 270532608, 410338674, 617285124, 893871740, 1290016512, 1801914272, 2513845188, 3404825448, 4624416768, 6103593750, 8094558822, 10465136172

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 7th powers of the divisor complements of the squarefree divisors of n.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A008683 (mu), A069092.

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), this sequence (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

f[p_, e_] := p^(7*e) + p^(7*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Feb 08 2022 *)

PARI
```
a(n)=sumdiv(n, d, moebius(n/d)^2*d^7);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^7*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

a(n) = Sum_{d|n} d^7 * mu(n/d)^2.
a(n) = n^7 * Sum_{d|n} mu(d)^2 / d^7.
Multiplicative with a(p^e) = p^(7*e) + p^(7*e-7). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^8 * zeta(8) / (8 * zeta(16)) = 34459425 * n^8 / (28936 * Pi^8).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^7/(p^14-1)) = 1.008287998838997802253937842472728682107868602338715231926150271159410... (End)
a(n) = J_14(n) / J_7(n) = J_14(n) / A069092(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 13 2022

A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

Original entry on oeis.org

1, 127, 1093, 8128, 19531, 138811, 137257, 520192, 796797, 2480437, 1948717, 8883904, 5229043, 17431639, 21347383, 33292288, 25646167, 101193219, 49659541, 158747968, 150021901, 247487059, 154764793, 568569856, 305171875, 664088461, 580865013, 1115624896

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^7 such that the quotient group Z^7 / L is C_n. - Álvar Ibeas, Oct 30 2015

Enrique Pérez Herrero and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..5000 from Enrique Pérez Herrero)
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 7 of A263950.

Cf. A000010, A000203, A008683, A013664, A013665, A069092.

A160897 := proc(n)
    add(numtheory[mobius](n/d)*d^7,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A160897(n)) ;
end do: # R. J. Mathar, Mar 14 2016

A160897[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(8 - 1)/EulerPhi[n] &] (* Enrique Pérez Herrero, Oct 27 2010 *)
f[p_, e_] := p^(6*e - 6) * (p^7-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(30, n, sumdiv(n^6, d, if(ispower(d, 7), moebius(sqrtnint(d, 7))*sigma(n^6/d), 0))) \\ Altug Alkan, Oct 30 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(6*f[i,2]-6)*(1+p+p^2+p^3+p^4+p^5+p^6); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_7(n)/J_1(n) = J_7(n)/phi(n) = A069092(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 27 2010
From Álvar Ibeas, Oct 30 2015: (Start)
Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).
For squarefree n, a(n) = A000203(n^6). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 + (p^6-1)/((p-1)*p^7)) = 0.2761554804... .
Sum_{k>=1} 1/a(k) = zeta(6)*zeta(7) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 1.008982290854... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^7). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 27 2010

A343509 a(n) = Sum_{k=1..n} gcd(k, n)^7.

Original entry on oeis.org

1, 129, 2189, 16514, 78129, 282381, 823549, 2113796, 4787349, 10078641, 19487181, 36149146, 62748529, 106237821, 171024381, 270565896, 410338689, 617568021, 893871757, 1290222306, 1802748761, 2513846349, 3404825469, 4627099444, 6103828145, 8094560241

Offset: 1

Seiichi Manyama, Apr 17 2021

In general, for m > 1, if a(n) = Sum_{j=1..n} gcd(j, n)^m, then Sum_{k=1..n} a(k) ~ zeta(m) * n^(m+1) / ((m+1) * zeta(m+1)). - Vaclav Kotesovec, May 20 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column 7 of A343510.

Cf. A000010, A013954 (sigma_6(n)), A069092, A343521.

a[n_] := Sum[GCD[k, n]^7, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^7);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^7);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 6));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+120*x^k+1191*x^(2*k)+2416*x^(3*k)+1191*x^(4*k)+120*x^(5*k)+x^(6*k))/(1-x^k)^8))

a(n) = Sum_{d|n} phi(n/d) * d^7.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_6(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8.
Dirichlet g.f.: zeta(s-1) * zeta(s-7) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 4725*zeta(7)*n^8 / (4*Pi^8). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_7 <= n} gcd(i_1, ..., i_7, n) = Sum_{d divides n} d * J_7(n/d), where the Jordan totient function J_7(n) = A069092(n). - Peter Bala, Jan 29 2024

A336488 Values taken by all the Jordan totient functions J_k(m) for k >= 1 and m >= 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 20, 22, 24, 26, 28, 30, 31, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 63, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 124, 126, 127, 128, 130, 132, 136, 138, 140, 144, 148

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

The asymptotic density of this sequence is 0 (Rao and Murty, 1979).

First differs from A221178 at n = 75, since a(75) = J_3(6) = 182 is not a term of A221178.

R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.

A002202 is a subsequence.
Cf. A000010, A007434, A059376, A059377, A059378, A059379, A059380, A069091, A069092, A069093, A069094, A069095, A221178.
Similar sequence: A211347.

phiQ[m_] := Select[Range[m + 1, 2 m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; jor[k_, n_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; jorval[k_, mx_] := jor[k, #] & /@ Range[Floor@Surd[mx*Zeta[k], k]]; mx = 300; Select[Union @ Flatten[{Select[Range[mx], phiQ], jorval[#, mx] & /@ Range[2, Floor[Log2[mx]]]}], # <= mx &] (* using code by Jean-François Alcover at A002202 *)

cp's OEIS Frontend

A000541 Sum of 7th powers: 1^7 + 2^7 + ... + n^7.

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A069091 Jordan function J_6(n).

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A351247 a(n) = n^7 * Sum_{p|n, p prime} 1/p^7.

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A351302 a(n) = n^7 * Product_{p|n, p prime} (1 + 1/p^7).

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A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

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A343509 a(n) = Sum_{k=1..n} gcd(k, n)^7.

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