A065958 -id:A065958 - cp's OEIS frontend

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

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

A351303 a(n) = n^8 * Product_{p|n, p prime} (1 + 1/p^8).

Original entry on oeis.org

1, 257, 6562, 65792, 390626, 1686434, 5764802, 16842752, 43053282, 100390882, 214358882, 431727104, 815730722, 1481554114, 2563287812, 4311744512, 6975757442, 11064693474, 16983563042, 25700065792, 37828630724, 55090232674, 78310985282, 110522138624, 152588281250

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 8th 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), this sequence (k=8), A351304 (k=9), A351305 (k=10).

f[p_, e_] := p^(8*e) + p^(8*(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^8);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^8*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

a(n) = Sum_{d|n} d^8 * mu(n/d)^2.
a(n) = n^8 * Sum_{d|n} mu(d)^2 / d^8.
Multiplicative with a(p^e) = p^(8*e) + p^(8*e-8). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^9 * zeta(9) / (9 * zeta(18)) = 4331032831125 * n^9 * zeta(9) / (43867 * Pi^18).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^8/(p^16-1)) = 1.004062071480173688638170669970682370243496458304179434830922739661777... (End)
a(n) = J_16(n)/J_8(n) = J_16(n)/A069093(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 14 2022

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

Original entry on oeis.org

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)

A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).

Original entry on oeis.org

1, 4, 9, 4, 25, 36, 49, 4, 9, 100, 121, 36, 169, 196, 225, 4, 289, 36, 361, 100, 441, 484, 529, 36, 25, 676, 9, 196, 841, 900, 961, 4, 1089, 1156, 1225, 36, 1369, 1444, 1521, 100, 1681, 1764, 1849, 484, 225, 2116, 2209, 36, 49, 100, 2601, 676, 2809, 36, 3025, 196

Offset: 1

Reinhard Zumkeller, Dec 10 2002

It is conjectured that only 1 and 1782 satisfy a(k) = sigma(k). See Broughan link. - Michel Marcus, Feb 28 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.
K. Broughan, J.-M. De Koninck, I. Kátai, and F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12.
Yong-Gao Chen, and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.

Cf. A002117, A007948, A055231, A062378, A330523, A007434, A008683, A001221, A065958.

a := n -> mul(f,f=map(x->x^2,select(isprime,divisors(n))));
seq(a(n), n=1..56);  # Peter Luschny, Mar 30 2014

a[n_] := Times @@ FactorInteger[n][[All, 1]]^2; Array[a, 60] (* Jean-François Alcover, Jun 04 2019 *)

a(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])^2 \\ Charles R Greathouse IV, Aug 06 2013

Multiplicative with a(p^e) = p^2. - Mitch Harris, May 17 2005
G.f.: Sum_{k>=1} mu(k)^2*J_2(k)*x^k/(1 - x^k), where J_2() is the Jordan function. - Ilya Gutkovskiy, Nov 06 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 / 3 = 0.214725... . - Amiram Eldar, Oct 30 2022
a(n) = Sum_{1 <= i, j <= n}  ( mobius(n/gcd(i, j, n)) )^2. - Peter Bala, Jan 28 2024
a(n) = Sum_{d|n} mu(d)^2*J_2(d), where J_2 = A007434. - Ridouane Oudra, Jul 24 2025
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*Psi_2(d), where omega = A001221 and Psi_2 = A065958. - Ridouane Oudra, Aug 01 2025

A328639 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-2)).

Original entry on oeis.org

1, -5, -10, 5, -26, 50, -50, -5, 10, 130, -122, -50, -170, 250, 260, 5, -290, -50, -362, -130, 500, 610, -530, 50, 26, 850, -10, -250, -842, -1300, -962, -5, 1220, 1450, 1300, 50, -1370, 1810, 1700, 130, -1682, -2500, -1850, -610, -260, 2650, -2210, -50, 50, -130

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A065958.

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

Cf. A008683, A008836, A026424 (positions of negative terms), A046970, A065958, A323363, A328640.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^2, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 50}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^2 &], {n, 1, 50}]
f[p_, e_] := (-1)^e*(p^2 + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA065958(n/d) * a(d).
a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^2, where lambda = A008836 and mu = A008683.
Multiplicative with a(p^e) = (-1)^e*(p^2 + 1). - Amiram Eldar, Nov 30 2020

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).

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)

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.

A321973 Partial sums of the Dedekind psi_2(k) function, for 1 <= k <= n.

Original entry on oeis.org

0, 1, 6, 16, 36, 62, 112, 162, 242, 332, 462, 584, 784, 954, 1204, 1464, 1784, 2074, 2524, 2886, 3406, 3906, 4516, 5046, 5846, 6496, 7346, 8156, 9156, 9998, 11298, 12260, 13540, 14760, 16210, 17510, 19310, 20680, 22490, 24190, 26270, 27952, 30452, 32302, 34742

Offset: 0

Daniel Suteu, Nov 22 2018

In general, for m >= 1, Sum_{k=1..n} psi_m(k) = Sum_{k=1..n} mu(k)^2 * (Bernoulli(m+1, 1+floor(n/k)) - Bernoulli(m+1, 1)) / (m+1), where mu(k) is the Moebius function and Bernoulli(n,x) are the Bernoulli polynomials.

In general, for m >= 1, Sum_{k=1..n} psi_m(k) ~ n^(m+1) * zeta(m+1) / ((m+1) * zeta(2*(m+1))).

Wikipedia, Dedekind psi function

Cf. A001615, A065958, A173290.

a[n_] := Sum[MoebiusMu[k]^2 * BernoulliB[3, 1 + Floor[n/k]], {k, 1, n}]/3; Array[a, 50, 0] (* Amiram Eldar, Nov 23 2018 *)

a(n) = sum(k=1, n, moebius(k)^2 * ((n\k)^3/3 + (n\k)^2/2 + (n\k)/6));

a(n) = Sum_{k=1..n} A065958(k).
a(n) ~ n^3 * zeta(3) / (3*zeta(6)).
a(n) = Sum_{k=1..n} mu(k)^2 * Bernoulli(3, 1+floor(n/k)) / 3.

cp's OEIS Frontend

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

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A351303 a(n) = n^8 * Product_{p|n, p prime} (1 + 1/p^8).

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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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A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).

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A328639 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-2)).

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

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A328639 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-2)).