A338547 -id:A338547

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A338548

a(n) = n^3 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^3.

+10
3

1, -9, 26, -56, 124, -234, 342, -448, 702, -1116, 1330, -1456, 2196, -3078, 3224, -3584, 4912, -6318, 6858, -6944, 8892, -11970, 12166, -11648, 15500, -19764, 18954, -19152, 24388, -29016, 29790, -28672, 34580, -44208, 42408, -39312, 50652, -61722, 57096, -55552, 68920, -80028

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - 4*x^k + x^(2*k)) / (1 + x^k)^4.

G.f. A(x) satisfies: A(x) = x * (1 - 4*x + x^2) / (1 + x)^4 - Sum_{k>=2} A(x^k).

Dirichlet g.f.: (1 - 2^(4 - s)) * zeta(s - 3) / zeta(s).

a(n) = J_3(n) if n odd, J_3(n) - 16 * J_3(n/2) if n even, where J_3 = A059376 (Jordan function J_3).

Multiplicative with a(2) = -9, a(2^e) = -7*2^(3*(e-1)) for e > 1, and a(p^e) = (p^3-1)*p^(3*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

MATHEMATICA

Table[n^3 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^3, {d, Divisors[n]}], {n, 1, 42}]

nmax = 42; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - 4 x^k + x^(2 k))/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

f[p_, e_] := (p^3 - 1)*p^(3*(e - 1)); f[2, 1] = -9; f[2, e_] := -7*2^(3*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

PROG

(PARI) a(n) = n^3 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^3); \\ Michel Marcus, Nov 02 2020

CROSSREFS

Cf. A008683, A059376, A138503, A325596, A338547, A338549.

KEYWORD

sign,mult

AUTHOR

Ilya Gutkovskiy, Nov 02 2020

STATUS

approved

A338549

a(n) = n^4 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^4.

+10
3

1, -17, 80, -240, 624, -1360, 2400, -3840, 6480, -10608, 14640, -19200, 28560, -40800, 49920, -61440, 83520, -110160, 130320, -149760, 192000, -248880, 279840, -307200, 390000, -485520, 524880, -576000, 707280, -848640, 923520, -983040, 1171200, -1419840, 1497600, -1555200

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - 11*x^k + 11*x^(2*k) - x^(3*k)) / (1 + x^k)^5.

G.f. A(x) satisfies: A(x) = x * (1 - 11*x + 11*x^2 - x^3) / (1 + x)^5 - Sum_{k>=2} A(x^k).

Dirichlet g.f.: (1 - 2^(5 - s)) * zeta(s - 4) / zeta(s).

a(n) = J_4(n) if n odd, J_4(n) - 32 * J_4(n/2) if n even, where J_4 = A059377 (Jordan function J_4).

Multiplicative with a(2) = -17, a(2^e) = -15*2^(4*(e-1)) for e > 1, and a(p^e) = (p^4-1)*p^(4*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

MATHEMATICA

Table[n^4 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^4, {d, Divisors[n]}], {n, 1, 36}]

nmax = 36; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - 11 x^k + 11 x^(2 k) - x^(3 k))/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

f[p_, e_] := (p^4 - 1)*p^(4*(e - 1)); f[2, 1] = -17; f[2, e_] := -15*2^(4*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)