A065959 -id:A065959 - cp's OEIS frontend

A351304 a(n) = n^9 * Product_{p|n, p prime} (1 + 1/p^9).

1, 513, 19684, 262656, 1953126, 10097892, 40353608, 134479872, 387440172, 1001953638, 2357947692, 5170120704, 10604499374, 20701400904, 38445332184, 68853694464, 118587876498, 198756808236, 322687697780, 513000262656, 794320419872, 1209627165996, 1801152661464, 2647101800448

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 9th 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).

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), A351302 (k=7), A351303 (k=8), this sequence (k=9), A351305 (k=10).

f[p_, e_] := p^(9*e) + p^(9*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Feb 08 2022 *)

PARI
```
a(n)=sumdiv(n, d, moebius(n/d)^2*d^9);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^9*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

from math import prod
from sympy import factorint
def A351304(n): return prod(p**(9*e)+p**(9*(e-1)) for p,e in factorint(n).items()) # Chai Wah Wu, Sep 28 2024

a(n) = Sum_{d|n} d^9 * mu(n/d)^2.
a(n) = n^9 * Sum_{d|n} mu(d)^2 / d^9.
Multiplicative with a(p^e) = p^(9*e) + p^(9*e-9). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^10 * zeta(10) / (10 * zeta(20)) = 3273645375 * n^10 / (349222 * Pi^10).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^9/(p^18-1)) = 1.002004575331916689985388864168116922608947780516939765639888137700557... (End)

A351305 a(n) = n^10 * Product_{p|n, p prime} (1 + 1/p^10).

Original entry on oeis.org

1, 1025, 59050, 1049600, 9765626, 60526250, 282475250, 1074790400, 3486843450, 10009766650, 25937424602, 61978880000, 137858491850, 289537131250, 576660215300, 1100585369600, 2015993900450, 3574014536250, 6131066257802, 10250001049600, 16680163512500, 26585860217050

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 10th 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).

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), A351302 (k=7), A351303 (k=8), A351304 (k=9), this sequence (k=10).

f[p_, e_] := p^(10*e) + p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Feb 08 2022 *)

a(n)=sumdiv(n, d, moebius(n/d)^2*d^10);

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^10*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

a(n) = Sum_{d|n} d^10 * mu(n/d)^2.
a(n) = n^10 * Sum_{d|n} mu(d)^2 / d^10.
Multiplicative with a(p^e) = p^(10*e) + p^(10*e-10). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-10)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^11 * zeta(11) / (11 * zeta(22)) = 1222532449149375 * n^11 * zeta(11) / (155366 * Pi^22).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^10/(p^20-1)) = 1.000993621149252443797467720671490169127513829380371486971107300011796... (End)

A194532 Jordan function ratio J_6(n)/J_2(n).

Original entry on oeis.org

1, 21, 91, 336, 651, 1911, 2451, 5376, 7371, 13671, 14763, 30576, 28731, 51471, 59241, 86016, 83811, 154791, 130683, 218736, 223041, 310023, 280371, 489216, 406875, 603351, 597051, 823536, 708123, 1244061, 924483, 1376256, 1343433, 1760031, 1595601, 2476656, 1875531, 2744343

Offset: 1

R. J. Mathar, Aug 28 2011

Dirichlet convolution of A000583 with the multiplicative function which starts 1, 5, 10, 0, 26, 50, 50, 0, 0, 130, 122, 0, 170, 250, 260, 0, 290,..

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007434, A065959, A069091.

f:= proc(n) local t;
     mul(t[1]^(4*(t[2]-1))*((t[1]^2+1)^2-t[1]^2),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jun 14 2016

JordanTotient[n_, k_: 1] := DivisorSum[n, #^k MoebiusMu[n/#] &] /; (n > 0) && IntegerQ@ n; Table[JordanTotient[n, 6]/JordanTotient[n, 2], {n, 12}] (* Michael De Vlieger, Jun 14 2016, after Enrique Pérez Herrero at A065959 *)
f[p_, e_] := p^(4*(e-1))*(p^2+p+1)*(p^2-p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(4*(f[i,2]-1))*(f[i,1]^2+f[i,1]+1)*(f[i,1]^2-f[i,1]+1));} \\ Amiram Eldar, Nov 05 2022

a(n) = A069091(n)/A007434(n).
Multiplicative with a(p^e) = p^(4*(e-1))*(p^2+p+1)*(p^2-p+1), e>0.
Dirichlet g.f.: zeta(s-4)*product_{primes p} (1+p^(2-s)+p^(-s)).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = Product_{primes p} (1 + 1/p^3 + 1/p^5) = 1.2196771388395597011492820972459808778277319864216893177353903924... - Vaclav Kotesovec, Dec 18 2019
Sum_{n>=1} 1/a(n) = (Pi^8/14175) * Product_{p prime} (1 + 1/p^2 + 1/p^4 - 1/p^6 - 1/p^8) = 1.06469274411... . - Amiram Eldar, Nov 05 2022

A328640 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-3)).

Original entry on oeis.org

1, -9, -28, 9, -126, 252, -344, -9, 28, 1134, -1332, -252, -2198, 3096, 3528, 9, -4914, -252, -6860, -1134, 9632, 11988, -12168, 252, 126, 19782, -28, -3096, -24390, -31752, -29792, -9, 37296, 44226, 43344, 252, -50654, 61740, 61544, 1134, -68922, -86688, -79508, -11988, -3528

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A065959.

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

Cf. A008683, A008836, A026424 (positions of negative terms), A063453, A065959, A323363, A328639.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^3, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^3 &], {n, 1, 45}]
f[p_, e_] := (-1)^e*(p^3+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^3)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA065959(n/d) * a(d).
a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^3, where lambda = A008836 and mu = A008683.
Multiplicative with a(p^) = (-1)^e*(p^3+1). - Amiram Eldar, Dec 03 2022

A320974 a(n) = n^n * Product_{p|n, p prime} (1 + 1/p^n).

Original entry on oeis.org

1, 5, 28, 272, 3126, 47450, 823544, 16842752, 387440172, 10009766650, 285311670612, 8918294011904, 302875106592254, 11112685048647250, 437893920912786408, 18447025548686262272, 827240261886336764178, 39346558271492178663450, 1978419655660313589123980

Offset: 1

Ilya Gutkovskiy, Oct 25 2018

nonn

Wikipedia, Dedekind psi function

Cf. A001615, A008683, A065958, A065959, A065960, A067858, A320973.

Table[n^n Product[1 + Boole[PrimeQ[d]]/d^n, {d, Divisors[n]}], {n, 19}]
Table[SeriesCoefficient[Sum[MoebiusMu[k]^2 PolyLog[-n, x^k], {k, 1, n}], {x, 0, n}], {n, 19}]
Table[Sum[MoebiusMu[n/d]^2 d^n, {d, Divisors[n]}], {n, 19}]

a(n) = [x^n] Sum_{k>=1} mu(k)^2*PolyLog(-n,x^k), where PolyLog() is the polylogarithm function.
a(n) = Sum_{d|n} mu(n/d)^2*d^n.
a(n) = A320973(n,n).

A320973 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 9, 10, 6, 2, 1, 17, 28, 20, 6, 4, 1, 33, 82, 72, 26, 12, 2, 1, 65, 244, 272, 126, 50, 8, 2, 1, 129, 730, 1056, 626, 252, 50, 12, 2, 1, 257, 2188, 4160, 3126, 1394, 344, 80, 12, 4, 1, 513, 6562, 16512, 15626, 8052, 2402, 576, 90, 18, 2

Offset: 1

Ilya Gutkovskiy, Oct 25 2018

			Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  2,   3,   5,    9,    17,    33,  ...
  2,   4,  10,   28,    82,   244,  ...
  2,   6,  20,   72,   272,  1056,  ...
  2,   6,  26,  126,   626,  3126,  ...
  4,  12,  50,  252,  1394,  8052,  ...

Wikipedia, Dedekind psi function

Columns k=0..4 give A034444, A001615, A065958, A065959, A065960.

Cf. A008683, A059379, A059380, A320974 (diagonal).

Table[Function[k, n^k Product[1 + Boole[PrimeQ[d]]/d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j]^2 PolyLog[-k, x^j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[n/d]^2 d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} mu(j)^2*PolyLog(-k,x^j), where PolyLog() is the polylogarithm function.

A(n,k) = Sum_{d|n} mu(n/d)^2*d^k.

cp's OEIS Frontend

A351304 a(n) = n^9 * Product_{p|n, p prime} (1 + 1/p^9).

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A351305 a(n) = n^10 * Product_{p|n, p prime} (1 + 1/p^10).

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A194532 Jordan function ratio J_6(n)/J_2(n).

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A328640 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-3)).

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A320974 a(n) = n^n * Product_{p|n, p prime} (1 + 1/p^n).

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A320973 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).

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A328640 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-3)).