A007427 -id:A007427 - cp's OEIS frontend

A053822 Dirichlet inverse of sigma_2 function (A001157).

1, -5, -10, 4, -26, 50, -50, 0, 9, 130, -122, -40, -170, 250, 260, 0, -290, -45, -362, -104, 500, 610, -530, 0, 25, 850, 0, -200, -842, -1300, -962, 0, 1220, 1450, 1300, 36, -1370, 1810, 1700, 0, -1682, -2500, -1850, -488, -234, 2650, -2210, 0, 49, -125, 2900, -680

Offset: 1

N. J. A. Sloane, Apr 08 2000

sigma_2(n) is the sum of the squares of the divisors of n (A001157).

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.

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

Dirichlet inverse of sigma_k(n): A007427 (k = 0), A046692 (k = 1), A053825 (k = 3), A053826 (k = 4), A178448 (k = 5).

Cf. A001157,.

f1:= proc(p,e) if e = 1 then -1-p^2 elif e=2 then p^2 else 0 fi end proc:
f:= n -> mul(f1(t[1],t[2]),t=ifactors(n)[2]);
map(f, [$1..100]); # Robert Israel, Jan 29 2018

a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^2, {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 05 2019, after Ilya Gutkovskiy *)
f[p_, e_] := If[e == 1, -p^2 - 1, If[e == 2, p^2, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 2)))} \\ Andrew Howroyd, Aug 05 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020

Dirichlet g.f.: 1/(zeta(s)*zeta(s-2)).
Multiplicative with a(p^1) = -1-p^2, a(p^2) = p^2, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005
a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^2. - Ilya Gutkovskiy, Nov 06 2018
From Peter Bala, Jan 26 2024: (Start)
a(n) = Sum_{d divides n} d * (sigma(d))^(-1) * phi(n/d), where (sigma(n))^(-1) = A046692(n) denotes the Dirichlet inverse of sigma(n) = A000203(n).
a(n) = Sum_{d divides n} d^2 * (sigma_k(d))^(-1) * J_(k+2)(n/d) for k >= 0, where (sigma_k(n))^(-1) denotes the Dirichlet inverse of the divisor sum function sigma_k(n) and J_k(n) denotes the Jordan totient function. (End)

A053826 Dirichlet inverse of sigma_4 function (A001159).

Original entry on oeis.org

1, -17, -82, 16, -626, 1394, -2402, 0, 81, 10642, -14642, -1312, -28562, 40834, 51332, 0, -83522, -1377, -130322, -10016, 196964, 248914, -279842, 0, 625, 485554, 0, -38432, -707282, -872644, -923522, 0, 1200644, 1419874, 1503652, 1296, -1874162

Offset: 1

N. J. A. Sloane, Apr 08 2000

sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Dirichlet inverse of sigma_k(n): A007427 (k = 0), A046692 (k = 1), A053822(k = 2), A053825 (k = 3), A178448 (k = 5).

Cf. A001159, A046099 (where a(n) = 0).

Table[DivisorSum[n, MoebiusMu[n/#]*MoebiusMu[#]*#^4  &], {n, 1, 50}] (* G. C. Greubel, Nov 07 2018 *)
f[p_, e_] := If[e == 1, -p^4 - 1, If[e == 2, p^4, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*d^4); \\ Michel Marcus, Nov 06 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^4*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020

Dirichlet g.f.: 1/(zeta(s)*zeta(s-4)).
Multiplicative with a(p^1) = -1 - p^4, a(p^2) = p^4, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005
a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^4. - Ilya Gutkovskiy, Nov 06 2018
From Peter Bala, Jan 17 2024: (Start)
a(n) = Sum_{d divides n} d * A053825(d) * phi(n/d), where the totient function phi(n) = A000010(n).
a(n) = Sum_{d divides n} d^2 * (sigma_2(d))^(-1) * J_2(n/d),
a(n) = Sum_{d divides n} d^3 * (sigma_1(d))^(-1) * J_3(n/d), and for k >= 0,
a(n) = Sum_{d divides n} d^4 * (sigma_k(d))^(-1) * J_(k+4)(n/d), where (sigma_k(n))^(-1) denotes the Dirichlet inverse of the divisor sum function sigma_k(n) and J_k(n) denotes the Jordan totient function. (End)

A247343 Moebius transform applied four times to sequence 1,0,0,0,....

Original entry on oeis.org

1, -4, -4, 6, -4, 16, -4, -4, 6, 16, -4, -24, -4, 16, 16, 1, -4, -24, -4, -24, 16, 16, -4, 16, 6, 16, -4, -24, -4, -64, -4, 0, 16, 16, 16, 36, -4, 16, 16, 16, -4, -64, -4, -24, -24, 16, -4, -4, 6, -24, 16, -24, -4, 16, 16, 16, 16, 16, -4, 96, -4, 16, -24, 0, 16, -64, -4, -24, 16, -64

Offset: 1

Enrique Pérez Herrero, Sep 14 2014

Multiplicative because the Moebius transform of a multiplicative sequence is multiplicative. - Andrew Howroyd, Jul 25 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Enrique Pérez Herrero)

Cf. A000007, A008683, A007427, A007428.

tau[1, n_Integer]:=1; SetAttributes[tau, Listable];
tau[k_Integer, n_Integer]:=Plus@@(tau[k-1, Divisors[n]])/; k > 1;
tau[k_Integer, n_Integer]:=Plus@@(tau[k+1, Divisors[n]]*MoebiusMu[n/Divisors[n]]); k<1;
Table[tau[-4, n], {n, 70}]
f[p_, e_] := (-1)^e * Binomial[4, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

seq(n)={my(v=vector(n, n, n==1)); for(k=1, 4, v=dirmul(v, vector(#v, n, moebius(n)))); v} \\ Andrew Howroyd, Jul 25 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Dirichlet g.f.: 1/zeta(s)^4.

Multiplicative with a(p^e) = (-1)^e * binomial(4, e). - Amiram Eldar, Sep 11 2020

A252505 Number of biquadratefree (4th power free) divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 8, 2, 6, 6, 9

Offset: 1

Geoffrey Critzer, Mar 21 2015

Equivalently, a(n) is the number of divisors of n that are in A046100.
a(n) is also the number of divisors d such that the greatest common square divisor of d and n/d is 1.
The number of divisors d of n such that gcd(d, n/d) is squarefree. - Amiram Eldar, Aug 25 2023

			a(16) = 4 because there are 4 divisors of 16 that are 4th power free: 1,2,4,8.
a(16) = 4 because there are 4 divisors d of 16 such that the greatest common square divisor of d and 16/d is 1: 1,2,8,16.

Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 37, Exercise 1.27.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A252505, Sequence Machine.
Eric Weisstein's World of Mathematics, Biquadratefree.

Cf. A046100 (biquadratefree numbers).
Cf. A034444 (squarefree divisors), A073184 (cubefree divisors).
Cf. A001620.
Cf. A000005, A058035, A307430.
Also obtained as a Dirichlet convolution of the following pairs: A034444 and A227291, A007427 and A286779, A008966 and A323308, A048691 and A363552, A271102 and A322327, A307445 and A370296, and A018892 and A378214 (conjectured).

Prepend[Table[Apply[Times, (FactorInteger[n][[All, 2]] /. x_ /; x > 3 -> 3) + 1], {n, 2, 100}], 1]

isA046100(n) = (n==1) || vecmax(factor(n)[, 2])<4;
a(n) = {d = divisors(n); sum(i=1, #d, isA046100(d[i]));} \\ Michel Marcus, Mar 22 2015

a(n) = vecprod(apply(x->min(x, 3) + 1, factor(n)[, 2])); \\ Amiram Eldar, Aug 25 2023

Dirichlet g.f.: zeta(s)^2/zeta(4*s).
Sum_{k=1..n} a(k) ~ 90*n/Pi^4 * (log(n) - 1 + 2*gamma - 360*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} mu(gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
Multiplicative with a(p^e) = min(e, 3) + 1. - Amiram Eldar, Sep 19 2020
From Antti Karttunen, May 14 2025: (Start)
Following formulas have been generated for this sequence by Sequence Machine:
a(n) = A000005(A058035(n)).
a(n) = Sum_{d|n} A307430(d).
a(n) = Sum_{d|n} A034444(d)*A227291(n/d).
a(n) = Sum_{d|n} A007427(d)*A286779(n/d).
a(n) = Sum_{d|n} A008966(d)*A323308(n/d).
a(n) = Sum_{d|n} A048691(d)*A363552(n/d).
a(n) = Sum_{d|n} A271102(d)*A322327(n/d).
a(n) = Sum_{d|n} A307445(d)*A370296(n/d).
a(n) = Sum_{d|n} A018892(d)*A378214(n/d). [Conjectured]
(End)

A327276 a(n) = Sum_{d|n, d odd} mu(d) * mu(n/d).

Original entry on oeis.org

1, -1, -2, 0, -2, 2, -2, 0, 1, 2, -2, 0, -2, 2, 4, 0, -2, -1, -2, 0, 4, 2, -2, 0, 1, 2, 0, 0, -2, -4, -2, 0, 4, 2, 4, 0, -2, 2, 4, 0, -2, -4, -2, 0, -2, 2, -2, 0, 1, -1, 4, 0, -2, 0, 4, 0, 4, 2, -2, 0, -2, 2, -2, 0, 4, -4, -2, 0, 4, -4, -2, 0, -2, 2, -2, 0, 4, -4, -2

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

Dirichlet inverse of A001227.

All terms are 0 or +/- a power of 2. - Robert Israel, Nov 26 2019

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

Cf. A001227, A007427, A008683, A068068, A209229, A327278.

[&+[MoebiusMu(d)*MoebiusMu(n div d): d in [a:a in Divisors(n)| IsOdd(a)]]:n in [1..80]]; // Marius A. Burtea, Sep 15 2019

f:= proc(n) local m, d;
  m:= n/2^padic:-ordp(n,2);
  add(numtheory:-mobius(d)*numtheory:-mobius(n/d), d = numtheory:-divisors(m))
end proc:
map(f, [$1..100]); # Robert Israel, Nov 26 2019

Table[DivisorSum[n, MoebiusMu[#] MoebiusMu[n/#] &, OddQ[#] &], {n, 1, 79}]
a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSum[n/d, Mod[#, 2] &] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 79}]
f[p_, e_] := Which[e == 1, -1 - Boole[p > 2], e == 2, Boole[p > 2], e > 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n)={sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d)))} \\ Andrew Howroyd, Sep 23 2019

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A001227(k) * A(x^k).
Dirichlet g.f.: 1 / (zeta(s)^2 * (1 - 1/2^s)).
a(1) = 1; a(n) = -Sum_{d|n, dA001227(n/d) * a(d).
a(n) = Sum_{d|n} A209229(n/d) * A007427(d).
Multiplicative with a(2^e) = -1 if e = 1, and 0 if e > 1, and a(p^e) = -2 if e = 1, 1 if e = 2, and 0 if e > 2, for an odd prime p. - Amiram Eldar, Oct 25 2020

A341831 Dirichlet g.f.: 1 / zeta(s)^5.

Original entry on oeis.org

1, -5, -5, 10, -5, 25, -5, -10, 10, 25, -5, -50, -5, 25, 25, 5, -5, -50, -5, -50, 25, 25, -5, 50, 10, 25, -10, -50, -5, -125, -5, -1, 25, 25, 25, 100, -5, 25, 25, 50, -5, -125, -5, -50, -50, 25, -5, -25, 10, -50, 25, -50, -5, 50, 25, 50, 25, 25, -5, 250, -5, 25, -50, 0, 25

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A061200.

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

Row 5 of A346148.

Cf. A007427, A007428, A008683, A061200, A247343, A341832, A341833, A341834, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[5, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 65}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(5, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_5(n/d) * a(d).

A341832 Dirichlet g.f.: 1 / zeta(s)^6.

Original entry on oeis.org

1, -6, -6, 15, -6, 36, -6, -20, 15, 36, -6, -90, -6, 36, 36, 15, -6, -90, -6, -90, 36, 36, -6, 120, 15, 36, -20, -90, -6, -216, -6, -6, 36, 36, 36, 225, -6, 36, 36, 120, -6, -216, -6, -90, -90, 36, -6, -90, 15, -90, 36, -90, -6, 120, 36, 120, 36, 36, -6, 540, -6, 36, -90, 1, 36

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A034695.

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

Row 6 of A346148.

Cf. A007427, A007428, A008683, A034695, A247343, A341831, A341833, A341834, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[6, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 65}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^6)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(6, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_6(n/d) * a(d).

A341833 Dirichlet g.f.: 1 / zeta(s)^7.

Original entry on oeis.org

1, -7, -7, 21, -7, 49, -7, -35, 21, 49, -7, -147, -7, 49, 49, 35, -7, -147, -7, -147, 49, 49, -7, 245, 21, 49, -35, -147, -7, -343, -7, -21, 49, 49, 49, 441, -7, 49, 49, 245, -7, -343, -7, -147, -147, 49, -7, -245, 21, -147, 49, -147, -7, 245, 49, 245, 49, 49, -7, 1029, -7, 49

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111217.

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

Row 7 of A346148.

Cf. A007427, A007428, A008683, A111217, A247343, A341831, A341832, A341834, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[7, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 62}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^7)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(7, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_7(n/d) * a(d).

A341834 Dirichlet g.f.: 1 / zeta(s)^8.

Original entry on oeis.org

1, -8, -8, 28, -8, 64, -8, -56, 28, 64, -8, -224, -8, 64, 64, 70, -8, -224, -8, -224, 64, 64, -8, 448, 28, 64, -56, -224, -8, -512, -8, -56, 64, 64, 64, 784, -8, 64, 64, 448, -8, -512, -8, -224, -224, 64, -8, -560, 28, -224, 64, -224, -8, 448, 64, 448, 64, 64, -8, 1792

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111218.

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

Row 8 of A346148.

Cf. A007427, A007428, A008683, A111218, A247343, A341831, A341832, A341833, A341835, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[8, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 60}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^8)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(8, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_8(n/d) * a(d).

A341835 Dirichlet g.f.: 1 / zeta(s)^9.

Original entry on oeis.org

1, -9, -9, 36, -9, 81, -9, -84, 36, 81, -9, -324, -9, 81, 81, 126, -9, -324, -9, -324, 81, 81, -9, 756, 36, 81, -84, -324, -9, -729, -9, -126, 81, 81, 81, 1296, -9, 81, 81, 756, -9, -729, -9, -324, -324, 81, -9, -1134, 36, -324, 81, -324, -9, 756, 81, 756, 81, 81

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111219.

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

Row 9 of A346148.

Cf. A007427, A007428, A008683, A111219, A247343, A341831, A341832, A341833, A341834, A341836.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[9, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 58}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^9)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(9, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_9(n/d) * a(d).

cp's OEIS Frontend

A053822 Dirichlet inverse of sigma_2 function (A001157).

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A053826 Dirichlet inverse of sigma_4 function (A001159).

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A247343 Moebius transform applied four times to sequence 1,0,0,0,....

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A252505 Number of biquadratefree (4th power free) divisors of n.

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A327276 a(n) = Sum_{d|n, d odd} mu(d) * mu(n/d).

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A341831 Dirichlet g.f.: 1 / zeta(s)^5.

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A341832 Dirichlet g.f.: 1 / zeta(s)^6.

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