A007428 -id:A007428 - cp's OEIS frontend

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

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).

A341836 Dirichlet g.f.: 1 / zeta(s)^10.

Original entry on oeis.org

1, -10, -10, 45, -10, 100, -10, -120, 45, 100, -10, -450, -10, 100, 100, 210, -10, -450, -10, -450, 100, 100, -10, 1200, 45, 100, -120, -450, -10, -1000, -10, -252, 100, 100, 100, 2025, -10, 100, 100, 1200, -10, -1000, -10, -450, -450, 100, -10, -2100, 45, -450, 100, -450

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111220.

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

Row 10 of A346148.

Cf. A007427, A007428, A008683, A111220, A247343, A341831, A341832, A341833, A341834, A341835.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[10, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 52}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^10)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(10, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_10(n/d) * a(d).

A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 0, -2, -3, 1, -1, 1, -3, -4, 1, 1, -2, 3, -4, -5, 1, -1, 4, -3, 6, -5, -6, 1, 0, -2, 9, -4, 10, -6, -7, 1, 0, 0, -3, 16, -5, 15, -7, -8, 1, 1, 1, -1, -4, 25, -6, 21, -8, -9, 1, -1, 4, 3, -4, -5, 36, -7, 28, -9, -10, 1, 0, -2, 9, 6, -10, -6

Offset: 1

Sebastian Karlsson, Aug 20 2021

			  n\k| 1    2    3    4    5    6    7    8    9   10   11    12 ...
  ---+--------------------------------------------------------------
   1 | 1   -1   -1    0   -1    1   -1    0    0    1   -1     0 ...
   2 | 1   -2   -2    1   -2    4   -2    0    1    4   -2    -2 ...
   3 | 1   -3   -3    3   -3    9   -3   -1    3    9   -3    -9 ...
   4 | 1   -4   -4    6   -4   16   -4   -4    6   16   -4   -24 ...
   5 | 1   -5   -5   10   -5   25   -5  -10   10   25   -5   -50 ...
   6 | 1   -6   -6   15   -6   36   -6  -20   15   36   -6   -90 ...
   7 | 1   -7   -7   21   -7   49   -7  -35   21   49   -7  -147 ...
   8 | 1   -8   -8   28   -8   64   -8  -56   28   64   -8  -224 ...
   9 | 1   -9   -9   36   -9   81   -9  -84   36   81   -9  -324 ...
  10 | 1  -10  -10   45  -10  100  -10 -120   45  100  -10  -450 ...
  11 | 1  -11  -11   55  -11  121  -11 -165   55  121  -11  -605 ...
  12 | 1  -12  -12   66  -12  144  -12 -220   66  144  -12  -792 ...
  13 | 1  -13  -13   78  -13  169  -13 -286   78  169  -13 -1014 ...
  14 | 1  -14  -14   91  -14  196  -14 -364   91  196  -14 -1274 ...
  15 | 1  -15  -15  105  -15  225  -15 -455  105  225  -15 -1575 ...
  ...

Sebastian Karlsson, Antidiagonals n = 1..140, flattened

Row n=1..10 give A008683, A007427, A007428, A247343, A341831, A341832, A341833, A341834, A341835, A341836.
Main diagonal gives A341837.
Cf. A077592, A163767,

T[n_, k_] := If[k == 1, 1, Product[(-1)^e Binomial[n, e], {e, FactorInteger[k][[All, 2]]}]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 13 2021 *)

T(n, k) = my(f=factor(k)); for (k=1, #f~, f[k,1] = binomial(n, f[k,2])*(-1)^f[k,2]; f[k,2]=1); factorback(f); \\ Michel Marcus, Aug 21 2021

from sympy import binomial, primefactors as pf, multiplicity as mult
from math import prod
def T(n, k):
    return prod((-1)**mult(p, k)*binomial(n, mult(p, k)) for p in pf(k))

If k = Product (p_j^m_j) then T(n, k) = Product (binomial(n, m_j)*(-1)^m_j).
Dirichlet g.f. of the n-th row: 1/zeta^n(s).
T(n, p) = -n.
T(n, n) = A341837(n).

A158523 Moebius transform of negated primes in factorization of n.

Original entry on oeis.org

1, -3, -4, 6, -6, 12, -8, -12, 12, 18, -12, -24, -14, 24, 24, 24, -18, -36, -20, -36, 32, 36, -24, 48, 30, 42, -36, -48, -30, -72, -32, -48, 48, 54, 48, 72, -38, 60, 56, 72, -42, -96, -44, -72, -72, 72, -48, -96, 56, -90, 72, -84, -54, 108, 72, 96, 80, 90, -60, 144, -62, 96, -96, 96, 84, -144, -68, -108, 96, -144

Offset: 1

Jaroslav Krizek, Mar 20 2009

			a(72) = a(2^3*3^2) = (-1)^3*(2+1)*2^(3-1) * (-1)^2*(3+1)*3^(2-1) = (-12)*12 = -144.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A061019, A008683, A061020, A007427, A000012, A007428, A000005, A001615, A001222, A000040, A006881, A120944, A000961, A378434.

f[p_, e_] := (-1)^e*(p + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 05 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i,2]*(f[i,1]+1)*f[i,1]^(f[i,2]-1));} \\ Amiram Eldar, Jan 05 2023

Multiplicative with a(p^e) = (-1)^e*(p+1)*p^(e-1), e>0. a(1)=1.
a(n) = mu(n) * A061019(n) = A008683(n) * A061019(n) = A061020(n) * A007427(n) = A061020(n) * A007428(n) * A000012(n) = A007427(n) * A000012(n) * A061019(n) = A007428(n) * A000005(n) * A061019(n), where operation * denotes Dirichlet convolution. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d).
Inverse Moebius transform gives A061019.
a(n) = (-1)^A001222(n)*A001615(n).
Apparently the Dirichlet inverse of A048250. - R. J. Mathar, Jul 15 2010
Dirichlet g.f.: zeta(2*s-2)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Jan 05 2023

More terms from Antti Karttunen, Nov 26 2024

A007432 Moebius transform applied thrice to natural numbers.

Original entry on oeis.org

1, -1, 0, 1, 2, 0, 4, 1, 3, -2, 8, 0, 10, -4, 0, 2, 14, -3, 16, 2, 0, -8, 20, 0, 13, -10, 8, 4, 26, 0, 28, 4, 0, -14, 8, 3, 34, -16, 0, 2, 38, 0, 40, 8, 6, -20, 44, 0, 31, -13, 0, 10, 50, -8, 16, 4, 0, -26, 56, 0, 58, -28, 12, 8, 20, 0, 64, 14

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (108*n^2/Pi^6) for n = 1..1000000
N. J. A. Sloane, Transforms

Cf. A007431.

a[p_, e_] := Sum[ (-1)^k*Binomial[3, k]*p^(e - k), {k, 0, Min[e, 3]}]; a[n_] := Times @@ Apply[a, FactorInteger[n], {1}]; a[1] = 1; Table[ a[n], {n, 1, 68}] (* Jean-François Alcover, Dec 28 2011, after formula *)

Multiplicative with a(p^e) = Sum_{k=0..3} (-1)^k C(3,k)*p^(e-k)[e>=k];
Dirichlet g.f.: zeta(s-1)/zeta^3(s).
a(n) = Sum{d|n} tau_{-3}(d)*n/d = Sum{d|n} tau_{-2}(d)*phi(n/d), where tau_{-3} is A007428 and tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
Sum_{k=1..n} a(k) ~ 108 * n^2 / Pi^6. - Vaclav Kotesovec, Nov 04 2018

A326415 Dirichlet g.f.: zeta(2*s) / zeta(s)^3.

Original entry on oeis.org

1, -3, -3, 4, -3, 9, -3, -4, 4, 9, -3, -12, -3, 9, 9, 4, -3, -12, -3, -12, 9, 9, -3, 12, 4, 9, -4, -12, -3, -27, -3, -4, 9, 9, 9, 16, -3, 9, 9, 12, -3, -27, -3, -12, -12, 9, -3, -12, 4, -12, 9, -12, -3, 12, 9, 12, 9, 9, -3, 36, -3, 9, -12, 4, 9, -27, -3, -12, 9, -27, -3, -16, -3, 9, -12

Offset: 1

Ilya Gutkovskiy, Oct 18 2019

Moebius transform applied twice to A008836.

Dirichlet inverse of A048691.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of partial sums of A326415 up to n=10000.

Cf. A000005, A001221, A001222, A007427, A007428, A008836, A010052, A034444, A046951, A048691, A158522.

Table[Sum[MoebiusMu[n/d] (-1)^PrimeOmega[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSigma[0, (n/d)^2] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 75}]
f[p_, e_] := If[e == 1, -3, (-1)^e*4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

a(n) = Sum_{d|n} mu(n/d) * (-1)^bigomega(d) * 2^omega(d), where mu = A008683, bigomega = A001222 and omega = A001221.
a(1) = 1; a(n) = -Sum_{d|n, dA000005.
a(n) = Sum_{d|n} A008836(n/d) * A007427(d).
a(n) = Sum_{d|n} A010052(n/d) * A007428(d).
Multiplicative with a(p^e) = -3 if e = 1, and 4*(-1)^e otherwise. - Amiram Eldar, Oct 26 2020
b(n) = abs( a(n) ) is multiplicative with b(p) = 3 and b(p^e) = 4 for e > 1 and prime p. Its Dirichlet g.f. is: zeta(s)^3 / zeta(2*s)^2. - Werner Schulte, Jan 18 2023

A326814 Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1 - 2 * p^(-s)).

Original entry on oeis.org

1, -3, -3, 2, -3, 9, -3, 0, 2, 9, -3, -6, -3, 9, 9, 0, -3, -6, -3, -6, 9, 9, -3, 0, 2, 9, 0, -6, -3, -27, -3, 0, 9, 9, 9, 4, -3, 9, 9, 0, -3, -27, -3, -6, -6, 9, -3, 0, 2, -6, 9, -6, -3, 0, 9, 0, 9, 9, -3, 18, -3, 9, -6, 0, 9, -27, -3, -6, 9, -27, -3, 0, -3, 9, -6

Offset: 1

Ilya Gutkovskiy, Oct 19 2019

Moebius transform applied twice to A076479 (unitary Moebius function).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of partial sums of A326814 up to n=10000.

Cf. A001221, A007428, A008683, A046099 (positions of 0's), A076479, A182139 (Dirichlet inverse), A226177, A326415, A326815.

Table[Sum[MoebiusMu[n/d] MoebiusMu[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Which[e == 1, -3, e == 2, 2, e > 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*2^omega(d)); \\ Michel Marcus, Oct 26 2020

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

a(n) = Sum_{d|n} mu(n/d) * mu(d) * 2^omega(d), where mu = A008683 and omega = A001221.

Multiplicative with a(p^e) = -3 if e = 1, 2 if e = 2, and 0 otherwise. - Amiram Eldar, Oct 26 2020

A341837 If n = Product (p_j^k_j) then a(n) = Product ((-1)^k_j * binomial(n, k_j)).

Original entry on oeis.org

1, -2, -3, 6, -5, 36, -7, -56, 36, 100, -11, -792, -13, 196, 225, 1820, -17, -2754, -19, -3800, 441, 484, -23, 48576, 300, 676, -2925, -10584, -29, -27000, -31, -201376, 1089, 1156, 1225, 396900, -37, 1444, 1521, 395200, -41, -74088, -43, -41624, -44550

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

sign

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

Cf. A007427, A007428, A008683, A163767, A247343, A341831, A341832, A341833, A341834, A341835, A341836.

Cf. A346148.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[n, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 45}]

a(n) = my(f=factor(n)[,2]); prod(k=1, #f, (-1)^f[k]*binomial(n, f[k])); \\ Michel Marcus, Feb 21 2021

a(n) = A346148(n, n). - Sebastian Karlsson, Aug 22 2021

A127173 T(n,k) = A007427(n/k) if k divides n, T(n,k) = 0 otherwise.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jan 06 2007

			First few rows of the triangle:
   1;
  -2,  1;
  -2,  0,  1;
   1, -2,  0,  1;
  -2,  0,  0,  0,  1;
   4, -2, -2,  0,  0, 1;
  -2,  0,  0,  0,  0, 0, 1;
   0,  1,  0, -2,  0, 0, 0, 1;
   1,  0, -2,  0,  0, 0, 0, 0, 1;
   4, -2,  0,  0, -2, 0, 0, 0, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A008683.
Column 1 is A007427.
Cf. A054525, A007431, A007428.

\\ here b(n) is A007427(n).
b(n)={sumdiv(n, d, moebius(d) * moebius(n/d))}
T(n,k)={if(n%k==0, b(n/k), 0)} \\ Andrew Howroyd, Feb 20 2022

Square of A054525 as lower triangular matrix.
A007431(n) = Sum_{k=1, n} k*T(n,k).
A007428(n) = Sum_{k=1..n} mu(k)*T(n,k).

Missing a(10)-a(14) and a(56) and beyond from Andrew Howroyd, Feb 20 2022

cp's OEIS Frontend

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

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

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A341836 Dirichlet g.f.: 1 / zeta(s)^10.

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A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

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A158523 Moebius transform of negated primes in factorization of n.

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A007432 Moebius transform applied thrice to natural numbers.

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A326415 Dirichlet g.f.: zeta(2*s) / zeta(s)^3.

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A326814 Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1 - 2 * p^(-s)).

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