A178448 -id:A178448 - cp's OEIS frontend

A001160 sigma_5(n), the sum of the 5th powers of the divisors of n.

1, 33, 244, 1057, 3126, 8052, 16808, 33825, 59293, 103158, 161052, 257908, 371294, 554664, 762744, 1082401, 1419858, 1956669, 2476100, 3304182, 4101152, 5314716, 6436344, 8253300, 9768751, 12252702, 14408200, 17766056, 20511150

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/504. - Simon Plouffe, Mar 01 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_6(z).

T. D. Noe, Table of n, a(n) for n = 1..10000
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001157, A001158, A001159, A013973, A000584 (Mobius transform), A178448 (Dirichlet inverse)

[DivisorSigma(5,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

A001160 := proc(n)
    numtheory[sigma][5](n);
end proc:
seq(A001160(n),n=1..30) ; # R. J. Mathar, Jan 31 2017

lst={};Do[AppendTo[lst,DivisorSigma[5,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[5,Range[30]] (* Harvey P. Dale, Nov 11 2013 *)

a(n)=sigma(n,5) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n, 5) for n in range(1, 30)]  # Zerinvary Lajos, Jun 04 2009

Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1). - David W. Wilson, Aug 01 2001
G.f.: sum(k>=1, k^5*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s)*zeta(s-5). - R. J. Mathar, Mar 06 2011
G.f. also (1 - E_6(q))/540, with the g.f. E_6 of A013973. See Hardy p. 166, (10.5.7) with R = E_6. - Wolfdieter Lang, Jan 31 2017
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^4)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
a(n) = Sum_{1 <= i, j, k, l, m <= n} tau(gcd(i, j, k, l, m, n)) = Sum_{d divides n} tau(d) * J_5(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 22 2024

A046692 Dirichlet inverse of sigma function (A000203).

Original entry on oeis.org

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

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)

A053822 Dirichlet inverse of sigma_2 function (A001157).

Original entry on oeis.org

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)

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

A001160 sigma_5(n), the sum of the 5th powers of the divisors of n.

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A046692 Dirichlet inverse of sigma function (A000203).

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A053825 Dirichlet inverse of sigma_3 function (A001158).

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A053822 Dirichlet inverse of sigma_2 function (A001157).

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A053826 Dirichlet inverse of sigma_4 function (A001159).

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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.

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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.