A053822 -id:A053822 - cp's OEIS frontend

A046692 Dirichlet inverse of sigma function (A000203).

1, -3, -4, 2, -6, 12, -8, 0, 3, 18, -12, -8, -14, 24, 24, 0, -18, -9, -20, -12, 32, 36, -24, 0, 5, 42, 0, -16, -30, -72, -32, 0, 48, 54, 48, 6, -38, 60, 56, 0, -42, -96, -44, -24, -18, 72, -48, 0, 7, -15, 72, -28, -54, 0, 72, 0, 80, 90, -60, 48, -62, 96, -24, 0, 84, -144, -68, -36, 96, -144, -72, 0, -74, 114, -20, -40, 96, -168

Offset: 1

Andrew R. Feist (arf22540(AT)cmsu2.cmsu.edu)

			a(36) = a(2^2*3^2) = 2*3 = 6.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
Andrew R. Feist, Fun With the Sigma-Function, unpub.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Cf. A000203, A053822, A053825, A053826, A178448.

t := 1; a := proc(n,t) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 then t1 := t1+d^t*mobius(d)*mobius(n/d); fi; od; t1; end;

a[n_] := (k = 0; Do[If[Mod[n, d] == 0, k = k + d*MoebiusMu[d]*MoebiusMu[n/d]], {d, 1, n}]; k); Table[a[n], {n, 1, 78}](* Jean-François Alcover, Oct 13 2011, after Maple *)
f[p_, e_] := Which[e == 1, -p-1, e == 2, p, e >= 3, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=if(n<1,0,direuler(p=2,n,(1-X)*(1-p*X))[n]) /* Ralf Stephan */

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

a(p) = -p-1, a(p^2) = p, a(p^k) = 0 for k > 2.
Dirichlet g.f.: 1/(zeta(s)*zeta(s-1)). - Benedict W. J. Irwin, Jul 10 2018
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} sigma(k)*A(x^k). - Ilya Gutkovskiy, May 11 2019
From Peter Bala, Jan 17 2024: (Start)
a(n) = Sum_{d divides n} d*mu(d)*mu(n/d). See Brown, p. 408.
a(n) = - Sum_{d divides n, d < n} a(d)*sigma_1(n/d).
a(n) = Sum_{d divides n} d*a(d)*J_2(n/d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{d divides n} d*A007427(d)*phi(n/d), where A007427 is the Dirichlet inverse of the tau function.
More generally, a(n) = Sum_{d divides n} d*sigma_[r]^(-1)(d)*J_(r+1)(n/d), where sigma_[r]^(-1) denotes the Dirichlet inverse of the function sigma_[r] = Sum_{d divides n} d^r.
a(n) = Sum_{k = 1..n} gcd(k, n)*A007427(gcd(k, n)).
a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)*a(gcd(j, k, n)). (End)
Sum_{k=1..n} abs(a(k)) ~ 45*n^2/Pi^4. - Vaclav Kotesovec, May 30 2024

Corrected by T. D. Noe, Nov 13 2006

A101035 Dirichlet inverse of the gcd-sum function (A018804).

Original entry on oeis.org

1, -3, -5, 1, -9, 15, -13, 1, 4, 27, -21, -5, -25, 39, 45, 1, -33, -12, -37, -9, 65, 63, -45, -5, 16, 75, 4, -13, -57, -135, -61, 1, 105, 99, 117, 4, -73, 111, 125, -9, -81, -195, -85, -21, -36, 135, -93, -5, 36, -48, 165, -25, -105, -12, 189, -13, 185, 171, -117, 45, -121, 183, -52, 1, 225, -315, -133, -33, 225, -351, -141, 4

Offset: 1

Gerard P. Michon, Nov 27 2004

			a(4)=1, a(8)=1, a(16)=1, a(32)=1, etc. because of the multiplicative definition for powers of 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. P. Michon, Multiplicative Functions.

Cf. A018804, A055615, A046692, A023900, A007427, A053822, A053825, A053826.

Cf. A008683.

a101035 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p 1 = 1 - 2 * p
   f p e = (p - 1) ^ 2
-- Reinhard Zumkeller, Jul 16 2012

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[ f[n/d]*DirichletInverse[f][d], {d, Most[ Divisors[n]]}]; GCDSum[n_] := Sum[ GCD[n, k], {k, 1, n}]; Table[ DirichletInverse[ GCDSum][n], {n, 1, 72}](* Jean-François Alcover, Dec 12 2011 *)
f[p_, e_] := If[e == 1, 1 - 2*p, (p - 1)^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 06 2022 *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv(n, d, n*eulerphi(d)/d)))} \\ Andrew Howroyd, Aug 05 2018

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

Multiplicative function with a(p)=1-2p and a(p^e)=(p-1)^2 when e>1 [p prime].
Dirichlet g.f.: zeta(s)/zeta^2(s-1). - R. J. Mathar, Apr 10 2011
a(n) = Sum{d|n} tau_{-2}(d)*d, where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
Conjecture: Logarithmic g.f. Sum_{n>0,k>0} mu(n)*mu(k)*log(1/(1-x^(n*k))). - Benedict W. J. Irwin, Jul 26 2017

A063453 Multiplicative with a(p^e) = 1 - p^3.

Original entry on oeis.org

1, -7, -26, -7, -124, 182, -342, -7, -26, 868, -1330, 182, -2196, 2394, 3224, -7, -4912, 182, -6858, 868, 8892, 9310, -12166, 182, -124, 15372, -26, 2394, -24388, -22568, -29790, -7, 34580, 34384, 42408, 182, -50652, 48006, 57096, 868, -68920, -62244, -79506, 9310, 3224, 85162, -103822, 182

Offset: 1

Vladeta Jovovic, Jul 26 2001

More generally, Dirichlet g.f. for Sum_{d|n} mu(d)*d^k, the Dirichlet inverse of the Jordan function J_k, is zeta(s)/zeta(s-k).
Apart from different signs also Sum_{d|n} core(d)^3*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002
Dirichlet inverse of A059376. - R. J. Mathar, Jul 15 2010

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A046970(r = 2), A189922 (r = 4).

Cf. A027748.

a063453 = product . map ((1 -) . (^ 3)) . a027748_row
-- Reinhard Zumkeller, Jan 19 2012

Jinvk := proc(n,k) local a,f,p ; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; a := a*(1-p^k) ; end do: a ; end proc:
A063453 := proc(n) Jinvk(n,3) ; end proc: # R. J. Mathar, Jul 04 2011
# second Maple program:
a:= n-> mul(1-i[1]^3, i=ifactors(n)[2]):
seq(a(n), n=1..48);  # Alois P. Heinz, Jan 26 2024

a[n_] := Total[MoebiusMu[#]*#^3& /@ Divisors[n]]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Jul 26 2011 *)
f[p_, e_] := (1-p^3); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)

a(n) = sumdiv(n, d, moebius(d) * d^3); \\ Indranil Ghosh, Mar 11 2017

from math import prod
from sympy import primefactors
def A063453(n): return prod(1-p**3 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

a(n) = Sum_{d|n} mu(d)*d^3.
Dirichlet g.f.: zeta(s)/zeta(s-3).
A023900(n) | a(n). - R. J. Mathar, Mar 30 2011
a(n)= product_{p|n}(1-p^3), n>=2, p prime, a(1)=1.  a(n)= J_{-3}(n)*n^3, with the Jordan function J_k(n). See the Apostol reference, p. 48, exercise 17. - Wolfdieter Lang, Jun 16 2011.
G.f.: Sum_{k>=1} mu(k)*k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
a(n) = Sum_{d divides n} d * sigma_2(d)^(-1) * sigma_1(n/d), where sigma_2(n)^(-1) = A053822(n) denotes the Dirichlet inverse of sigma_2(n). - Peter Bala, Jan 26 2024

A053825 Dirichlet inverse of sigma_3 function (A001158).

Original entry on oeis.org

1, -9, -28, 8, -126, 252, -344, 0, 27, 1134, -1332, -224, -2198, 3096, 3528, 0, -4914, -243, -6860, -1008, 9632, 11988, -12168, 0, 125, 19782, 0, -2752, -24390, -31752, -29792, 0, 37296, 44226, 43344, 216, -50654, 61740, 61544, 0, -68922, -86688

Offset: 1

N. J. A. Sloane, Apr 08 2000

sigma_3(n) is the sum of the cubes of the divisors of n (A001158).

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Cf. A001158, A046692, A053822, A053826, A178448.

with(numtheory):seq(add( mobius(n/d)*mobius(d)*d^3, d in divisors(n)), n = 1..100); # Peter Bala, Jan 26 2024

a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^3, {d, Divisors[n]}];
Array[a, 42] (* Jean-François Alcover, Aug 16 2019, after Ilya Gutkovskiy *)
f[p_, e_] := If[e == 1, -p^3 - 1, If[e == 2, p^3, 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, 3)))} \\ Andrew Howroyd, Aug 05 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^3*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020

Dirichlet g.f.: 1/(zeta(s)*zeta(s-3))
Multiplicative with a(p^1) = -1-p^3, a(p^2) = p^3, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005
a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^3. - Ilya Gutkovskiy, Nov 06 2018
From Peter Bala, Jan 17 2024: (Start)
a(n) = Sum_{d divides n} d * A053822(d) * phi(n/d), where the totient function phi(n) = A000010(n).
a(n) = Sum_{d divides n} d^2 * (sigma_1(d))^(-1) * J_2(n/d) and
a(n) = Sum_{d divides n} d^3 * (sigma_k(d))^(-1) * J_(k+3)(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)

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)

A334657 Dirichlet g.f.: 1 / zeta(s-2).

Original entry on oeis.org

1, -4, -9, 0, -25, 36, -49, 0, 0, 100, -121, 0, -169, 196, 225, 0, -289, 0, -361, 0, 441, 484, -529, 0, 0, 676, 0, 0, -841, -900, -961, 0, 1089, 1156, 1225, 0, -1369, 1444, 1521, 0, -1681, -1764, -1849, 0, 0, 2116, -2209, 0, 0, 0, 2601, 0, -2809, 0, 3025, 0, 3249, 3364, -3481, 0, -3721

Offset: 1

Ilya Gutkovskiy, May 07 2020

Dirichlet inverse of A000290.
Moebius transform of A046970.
Inverse Moebius transform of A053822.

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

Cf. A008683, A055615, A334659, A334660.

Cf. A000290, A046970, A053822.

Mathematica
```
Table[MoebiusMu[n] n^2, {n, 61}]
```

G.f. A(x) satisfies: A(x) = x - 2^2 * A(x^2) - 3^2 * A(x^3) - 4^2 * A(x^4) - ...
a(1) = 1; a(n) = -n^2 * Sum_{d|n, d < n} a(d) / d^2.
a(n) = mu(n) * n^2.
Multiplicative with a(p^e) = -p^2 if e = 1 and 0 otherwise. - Amiram Eldar, Oct 25 2020

A120630 Dirichlet inverse of A002654.

Original entry on oeis.org

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

Offset: 1

Gerard P. Michon, Jun 25 2006

			a(65)=4 because 65 is 5 times 13 and both of those primes are congruent to 1 modulo 4. Doubling an odd index yields the opposite of the value (e.g., a(130)=-4) because a(2)=-1. Doubling an even index yields zero.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002654, A006752, A023900, A046692, A053822, A053825, A053826, A101035.

A120630 := proc(n)
    local a,pp;
    if n = 1 then
        1;
    else
        a := 1 ;
        for pp in ifactors(n)[2] do
            if op(2,pp) > 2 then
                a := 0;
            elif op(1,pp) = 2 then
                if op(2,pp) = 1 then
                    a := -a ;
                else
                    a := 0 ;
                end if;
            elif modp(op(1,pp),4) = 3 then
                if op(2,pp) = 1 then
                    a := 0 ;
                else
                    a := -a ;
                end if;
            else
                if op(2,pp) = 1 then
                    a := -2*a ;
                else
                    ;
                end if;
            end if;
        end do:
        a;
    end if;
end proc: # R. J. Mathar, Sep 15 2015

A120630[n_] := Module[{a, pp}, If[n == 1, 1, a = 1; Do[Which[pp[[2]] > 2, a = 0, pp[[1]] == 2, If[pp[[2]] == 1, a = -a, a = 0], Mod[pp[[1]], 4] == 3, If[pp[[2]] == 1, a = 0, a = -a], True, If[pp[[2]] == 1, a = -2*a]], {pp, FactorInteger[n]}]; a]]; Array[A120630, 120] (* Jean-François Alcover, Apr 24 2017, after R. J. Mathar *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv( n, d, (d%4==1) - (d%4==3))))} \\ Andrew Howroyd, Aug 05 2018

Multiplicative function with a(p^e)=0 if e>2. a(2)=-1, a(4)=0. If p is a prime congruent to 3 modulo 4, then a(p)=0 and a(p^2)=-1. If p is a prime congruent to 1 modulo 4, then a(p)=-2 and a(p^2)=1.

Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 3/(2*Pi*G) = 0.521269..., and G is Catalan's constant (A006752). - Amiram Eldar, Jan 22 2024

A347227 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} mu(d)mu(n/d)d^k.

Original entry on oeis.org

1, 1, -2, 1, -3, -2, 1, -5, -4, 1, 1, -9, -10, 2, -2, 1, -17, -28, 4, -6, 4, 1, -33, -82, 8, -26, 12, -2, 1, -65, -244, 16, -126, 50, -8, 0, 1, -129, -730, 32, -626, 252, -50, 0, 1, 1, -257, -2188, 64, -3126, 1394, -344, 0, 3, 4, 1, -513, -6562, 128, -15626, 8052, -2402, 0, 9, 18, -2

Offset: 1

Seiichi Manyama, Aug 24 2021

			Square array begins:
   1,  1,   1,    1,    1,     1, ...
  -2, -3,  -5,   -9,  -17,   -33, ...
  -2, -4, -10,  -28,  -82,  -244, ...
   1,  2,   4,    8,   16,    32, ...
  -2, -6, -26, -126, -626, -3126, ...
   4, 12,  50,  252, 1394,  8052, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..5 give A007427, A046692, A053822, A053825, A053826, A178448.

T(n,n) gives A347251.

T[n_, k_] := DivisorSum[n, MoebiusMu[#] * MoebiusMu[n/#] * #^k &]; Table[T[n - k + 1, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Aug 24 2021 *)

T(n, k) = sumdiv(n, d, moebius(d)*moebius(n/d)*d^k);

Dirichlet g.f. of column k: 1/(zeta(s)*zeta(s-k)).

cp's OEIS Frontend

A046692 Dirichlet inverse of sigma function (A000203).

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A101035 Dirichlet inverse of the gcd-sum function (A018804).

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A063453 Multiplicative with a(p^e) = 1 - p^3.

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A053825 Dirichlet inverse of sigma_3 function (A001158).

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A053826 Dirichlet inverse of sigma_4 function (A001159).

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A334657 Dirichlet g.f.: 1 / zeta(s-2).

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A347227 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} mu(d)mu(n/d)d^k.