A069094 -id:A069094 - cp's OEIS frontend

A069095 Jordan function J_10(n).

1, 1023, 59048, 1047552, 9765624, 60406104, 282475248, 1072693248, 3486725352, 9990233352, 25937424600, 61855850496, 137858491848, 288972178704, 576640565952, 1098437885952, 2015993900448, 3566920035096, 6131066257800

Offset: 1

Benoit Cloitre, Apr 05 2002

a(n) is divisible by 264 = (2^3)*3*11 = A006863(5), except for n = 1, 2, 3 or 11. See Lugo. - Peter Bala, Jan 13 2024

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

Robert Israel, Table of n, a(n) for n = 1..10000
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), A069091 - A069094 (J_6 through J_9).

Cf. A013669.

f:= n -> n^10*mul(1-1/p^10, p=numtheory:-factorset(n)):
map(f, [$1..30]); # Robert Israel, Jan 09 2015

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 10]; Array[f, 21]
f[p_, e_] := p^(10*e) - p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PARI
```
a(n) = sumdiv(n,d,d^10*moebius(n/d));
```

a(n) = Sum_{d|n} d^10*mu(n/d).
Multiplicative with a(p^e) = p^(10e)-p^(10(e-1)).
Dirichlet generating function: zeta(s-10)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^10*Product_{distinct primes p dividing n} (1-1/p^10). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^11 / (11*zeta(11)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^10 = 1/zeta(11).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^10/(p^10-1)^2) = 1.0009955309... (End)

A007487 Sum of 9th powers.

Original entry on oeis.org

0, 1, 513, 20196, 282340, 2235465, 12313161, 52666768, 186884496, 574304985, 1574304985, 3932252676, 9092033028, 19696532401, 40357579185, 78800938560, 147520415296, 266108291793, 464467582161, 787155279940, 1299155279940, 2093435326521, 3300704544313

Offset: 0

N. J. A. Sloane

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.
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 Formula lines (second formula): 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.
Eric Weisstein's World of Mathematics, Power Sum.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Row 9 of array A103438.

Cf. A000217, A000542, A069094.

[&+[n^9: n in [0..m]]: m in [0..22]]; // Bruno Berselli, Aug 23 2011

[seq(add(i^9,i=1..n),n=0..40)];
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^9 od: seq(a[n], n=0..22); # Zerinvary Lajos, Feb 22 2008

lst={};s=0;Do[s=s+n^9;AppendTo[lst, s], {n, 10^2}];lst..or..Table[Sum[k^9, {k, 1, n}], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,30]^9] (* Harvey P. Dale, Oct 09 2016 *)

a(n)=n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 \\ Charles R Greathouse IV, Oct 07 2015

A007487_list, m = [0], [362880, -1451520, 2328480, -1905120, 834120, -186480, 18150, -510, 1, 0, 0]
for _ in range(10**2):
    for i in range(10):
        m[i+1]+= m[i]
    A007487_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

a(n) = n^2*(n+1)^2*(n^2+n-1)*(2*n^4+4*n^3-n^2-3*n+3)/20 (see MathWorld, Power Sum, formula 39). a(n) = n*A000542(n) - Sum_{i=0..n-1} A000542(i). - Bruno Berselli, Apr 26 2010
G.f.: x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1-x)^11. a(n) = a(-n-1). - Bruno Berselli, Aug 23 2011
a(n) = -Sum_{j=1..9} j*Stirling1(n+1,n+1-j)*Stirling2(n+9-j,n). - Mircea Merca, Jan 25 2014
a(n) = (16/5)*A000217(n)^5 - 4*A000217(n)^4 + (12/5)*A000217(n)^3 - (3/5)*A000217(n)^2. - Michael Raney, Mar 14 2016
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n > 10. - Wesley Ivan Hurt, Dec 21 2016
a(n) = 288*A005585(n-1)^2 + 1728*A108679(n-3) + A062392(n)^2. - Yasser Arath Chavez Reyes, May 11 2024
a(n) = Sum_{i=1..n} J_9(i)*floor(n/i), where J_9 is A069094. - Ridouane Oudra, Jul 17 2025

A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.

Original entry on oeis.org

0, 1, 1, 512, 1, 20195, 1, 262144, 19683, 1953637, 1, 10339840, 1, 40354119, 1972808, 134217728, 1, 397498185, 1, 1000262144, 40373290, 2357948203, 1, 5293998080, 1953125, 10604499885, 387420489, 20661308928, 1, 39453437071, 1, 68719476736, 2357967374, 118587877009, 42306732

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^9. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 20195; a(6) = 6^9 * Sum_{p|6, p prime} 1/p^9 = 10077696 * (1/2^9 + 1/3^9) = 20195.

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), A351247 (k=7), A351248 (k=8), this sequence (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069094.

Array[#^9*DivisorSum[#, 1/#^9 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

from sympy import primefactors
def A351249(n): return sum((n//p)**9 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069094(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)^9, where c = A010051.
a(p^k) = p^(9*k-9) for p prime and k>=1. (End)

A069092 Jordan function J_7(n).

Original entry on oeis.org

1, 127, 2186, 16256, 78124, 277622, 823542, 2080768, 4780782, 9921748, 19487170, 35535616, 62748516, 104589834, 170779064, 266338304, 410338672, 607159314, 893871738, 1269983744, 1800262812, 2474870590, 3404825446, 4548558848

Offset: 1

Benoit Cloitre, Apr 05 2002

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

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A069091 (J_6), A069092 (J_7), A069093 (J_8), A069094 (J_9), A069095 (J_10). [Enrique Pérez Herrero, Nov 02 2010]
Cf. A013666.

JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
A069092[n_] := JordanTotient[n, 7]; (* Enrique Pérez Herrero, Nov 02 2010 *)
f[p_, e_] := p^(7*e) - p^(7*(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^7*moebius(n/d)), ", "))

a(n) = Sum_{d|n} d^7*mu(n/d).
Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).
Dirichlet generating function: zeta(s-7)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A_7(x^n)/(1 - x^n)^8 = x + 127*x^2 + 2186*x^3 + 16256*x^4 + 78124*x^5 + ..., where A_7(x) = x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7 is the 7th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A160953 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.

Original entry on oeis.org

1, 511, 9841, 130816, 488281, 5028751, 6725601, 33488896, 64566801, 249511591, 235794769, 1287360256, 883708281, 3436782111, 4805173321, 8573157376, 7411742281, 32993635311, 17927094321, 63874967296, 66186639441, 120491126959, 81870575521, 329564225536

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^9 such that the quotient group Z^9 / L is C_n. - Álvar Ibeas, Nov 03 2015

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Enrique Pérez Herrero, Mathematica Package: Jordan Totient Function.
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 9 of A263950.

Cf. A000010, A000203, A008683, A013666, A013667, A069094.

A160953 := proc(n)
    add(numtheory[mobius](n/d)*d^9,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A160953(n)) ;
end do: # R. J. Mathar, Mar 14 2016

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A160953[n_]:=JordanTotient[n,9]/JordanTotient[n];
f[p_, e_] := p^(8*e - 8) * (p^9-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^8, d, if(ispower(d, 9), moebius(sqrtnint(d, 9))*sigma(n^8/d), 0))) \\ Altug Alkan, Nov 05 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; f[i,1] = p^(8*f[i,2]-8)*(p^9-1)/(p-1); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_9(n)/phi(n) = A069094(n)/A000010(n).
From Álvar Ibeas, Nov 03 2015: (Start)
Multiplicative with a(p^e) = p^(8e-8) * (p^9-1) / (p-1).
For squarefree n, a(n) = A000203(n^8). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 + (p^8-1)/((p-1)*p^9)) = 0.2156692448... .
Sum_{k>=1} 1/a(k) = zeta(8)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^17) = 1.002068659133... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^9). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.

Original entry on oeis.org

1, 73, 757, 4672, 15751, 55261, 117993, 299008, 551853, 1149823, 1772893, 3536704, 4829007, 8613489, 11923507, 19136512, 24142483, 40285269, 47052741, 73588672, 89320701, 129421189, 148048057, 226349056, 246109375, 352517511, 402300837, 551263296

Offset: 1

Seiichi Manyama, Mar 05 2025

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

Cf. A065958, A160889, A194532.
Cf. A059376, A069094.
Cf. A013664, A013667.

f[p_, e_] := p^(6*e) * (1 + 1/p^3 + 1/p^6); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 28] (* Amiram Eldar, Mar 05 2025 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n) = J(n, 9)/J(n, 3);

a(n) = {my(p = factor(n)[, 1]); n^6 * prod(i = 1, #p, 1 + 1/p[i]^3 + 1/p[i]^6);} \\ Amiram Eldar, Mar 05 2025

a(n) = A069094(n)/A059376(n).
a(n) = n^6 * Product_{distinct primes p dividing n} (1 + 1/p^3 + 1/p^6).
From Amiram Eldar, Mar 05 2025: (Start)
Dirichlet g.f.: zeta(s-6) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = Product_{p prime} (1 + 1/p^4 + 1/p^7) = 1.08635980686198102055... .
Sum_{n>=1} 1/a(n) = zeta(6)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^15) = 1.01533121878447451064... . (End)

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

A069095 Jordan function J_10(n).

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A007487 Sum of 9th powers.

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A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.

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A069092 Jordan function J_7(n).

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A160953 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.

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A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.

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

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