A189922 -id:A189922 - cp's OEIS frontend

A046970 Dirichlet inverse of the Jordan function J_2 (A007434).

1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576

Offset: 1

Douglas Stoll, dougstoll(AT)email.msn.com

B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2).

Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002

			a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3.
E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - _Jon Perry_, Aug 24 2010
G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ...

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.

Reinhard Zumkeller, 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].
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
David M. Smith, Multiple-Precision Evaluation of Bernoulli Numbers, Loyola Marymount Univ., 2024. See p. 3.
Wikipedia, Riemann zeta function.

Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A063453(r = 3), A189922 (r = 4).

Cf. A007434, A027641, A027642, A027748, A046692, A076479, A084920, A322360.

a046970 = product . map ((1 -) . (^ 2)) . 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:
A046970 := proc(n) Jinvk(n,2) ; end proc: # R. J. Mathar, Jul 04 2011

muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(1 - x[[i]][[1]]^2)]; p}, {n, 1, 50, 1}]] (* Jon Perry, Aug 24 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]]; (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)]; (* Michael Somos, Jan 11 2014 *)

A046970(n)=sumdiv(n,d,d^2*moebius(d)) \\ Benoit Cloitre

{a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])}; /* Michael Somos, Jan 11 2014 */

from math import prod
from sympy import primefactors
def A046970(n): return prod(1-p**2 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

Multiplicative with a(p^e) = 1 - p^2.
a(n) = Sum_{d|n} mu(d)*d^2.
abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010
From Wolfdieter Lang, Jun 16 2011: (Start)
Dirichlet g.f.: zeta(s)/zeta(s-2).
a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End)
a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011
G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
a(n) = Sum_{d divides n} d * (sigma_1(d))^(-1) * sigma_1(n/d), where (sigma_1(n))^(-1) = A046692(n) denotes the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024
a(n) = A076479(n) * A322360(n). - Amiram Eldar, Feb 02 2024

Corrected and extended by Vladeta Jovovic, Jul 25 2001

Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005

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

A321222 a(n) = Sum_{d|n} mu(d)*d^n.

Original entry on oeis.org

1, -3, -26, -15, -3124, 45864, -823542, -255, -19682, 9990233352, -285311670610, 2176246800, -302875106592252, 11111328602468784, 437893859848932344, -65535, -827240261886336764176, 101559568985784, -1978419655660313589123978, 99999904632567310800

Offset: 1

Ilya Gutkovskiy, Nov 06 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A008683, A023900, A046970, A063453, A067858, A189922, A189923.

Table[Sum[MoebiusMu[d] d^n, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[MoebiusMu[k] (k x)^k/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Product[1 - Boole[PrimeQ[d]] d^n, {d, Divisors[n]}], {n, 20}]

a(n) = sumdiv(n, d, moebius(d)*d^n) \\ Andrew Howroyd, Nov 06 2018

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

a(n) = Product_{p|n, p prime} (1 - p^n).

A334660 Dirichlet g.f.: 1 / zeta(s-4).

Original entry on oeis.org

1, -16, -81, 0, -625, 1296, -2401, 0, 0, 10000, -14641, 0, -28561, 38416, 50625, 0, -83521, 0, -130321, 0, 194481, 234256, -279841, 0, 0, 456976, 0, 0, -707281, -810000, -923521, 0, 1185921, 1336336, 1500625, 0, -1874161, 2085136, 2313441, 0, -2825761, -3111696, -3418801, 0, 0, 4477456

Offset: 1

Ilya Gutkovskiy, May 07 2020

Dirichlet inverse of A000583.
Moebius transform of A189922.
Inverse Moebius transform of A053826.

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

Cf. A008683, A055615, A334657, A334659.

Cf. A000583, A053826, A189922.

Mathematica
```
Table[MoebiusMu[n] n^4, {n, 46}]
```

G.f. A(x) satisfies: A(x) = x - 2^4 * A(x^2) - 3^4 * A(x^3) - 4^4 * A(x^4) - ...
a(1) = 1; a(n) = -n^4 * Sum_{d|n, d < n} a(d) / d^4.
a(n) = mu(n) * n^4.
Multiplicative with a(p^e) = -p^4 if e = 1 and 0 otherwise. - Amiram Eldar, Dec 05 2022

A189923 Jordan function J_{-5}(n) multiplied by n^5.

Original entry on oeis.org

1, -31, -242, -31, -3124, 7502, -16806, -31, -242, 96844, -161050, 7502, -371292, 520986, 756008, -31, -1419856, 7502, -2476098, 96844, 4067052, 4992550, -6436342, 7502, -3124, 11510052, -242, 520986, -20511148, -23436248

Offset: 1

Wolfdieter Lang, Jun 16 2011

For the Jordan function J_k see the Comtet and Apostol references.

			a(2) = a(4) = a(8) = ... = 1 - 2^5 = -31.
a(4) = mu(1)*1^5 + mu(2)*2^5 + mu(4)*4^5 = 1 - 32 + 0 = -31.
Sum identity for n=4: a(1)*(4/1)^5 + a(2)*(4/2)^5 + a(4)*(4/4)^5 = 1024 - 31*32 - 31 = 1.

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..200 from Indranil Ghosh)

Cf. A023900, A046970, A063453, A189922, for k=-1..-4.

a[n_] := Sum[ MoebiusMu[d]*d^5, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := (1-p^5); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)

for(n=1, 200, print1(sumdiv(n, d, moebius(d) * d^5),", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = sumdiv(n, d, moebius(d) * d^5); \\ Michel Marcus, Jan 14 2018

a(n) = J_{-5}(n)*n^5 = Product_{p prime |n} (1-p^5), for n>=2, a(1)=1.
a(n) = Sum_{d|n} mu(d)*d^5 with the Moebius function mu = A008683.
Dirichlet g.f.: zeta(s)/zeta(s-5).
Sum identity: Sum_{d|n} a(n)*(n/d)^5 = 1 for all n>=1.
a(n) = a(rad(n)) with rad(n) = A007947(n), the squarefree kernel of n.
G.f.: Sum_{k>=1} mu(k)*k^5*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017

A322324 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Product_{p|n, p prime} (1 - p^k).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -3, -2, 0, 1, -7, -8, -1, 0, 1, -15, -26, -3, -4, 0, 1, -31, -80, -7, -24, 2, 0, 1, -63, -242, -15, -124, 24, -6, 0, 1, -127, -728, -31, -624, 182, -48, -1, 0, 1, -255, -2186, -63, -3124, 1200, -342, -3, -2, 0, 1, -511, -6560, -127, -15624, 7502, -2400, -7, -8, 4, 0

Offset: 1

Ilya Gutkovskiy, Dec 03 2018

			Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -2,  -8,  -26,   -80,  -242, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -4, -24, -124,  -624, -3124, ...
  0,  2,  24,  182,  1200,  7502, ...

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

Columns k=0..5 give A063524, A023900, A046970, A063453, A189922, A189923.

Cf. A008683, A059379, A059380, A321222 (diagonal).

Table[Function[k, Product[1 - Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j] j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[d] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

T(n, k) = sumdiv(n, d, moebius(d)*d^k);
matrix(6, 6, n, k, T(n, k-1)) \\ Michel Marcus, Dec 03 2018

G.f. of column k: Sum_{j>=1} mu(j)*j^k*x^j/(1 - x^j).
Dirichlet g.f. of column k: zeta(s)/zeta(s-k).
A(n,k) = Sum_{d|n} mu(d)*d^k.

A192000 Sum of binomial numbers A000332(k+3), with k in the reduced residue system modulo n.

Original entry on oeis.org

0, 1, 6, 16, 56, 71, 252, 296, 651, 721, 2002, 1282, 4368, 3402, 5782, 6672, 15504, 7947, 26334, 15702, 28868, 28457, 65780, 30212, 85580, 63063, 103284, 81452, 201376, 66102, 278256, 174624, 255794, 228684, 383166, 206838, 658008, 391419, 576394, 413244, 1086008

Offset: 1

Wolfdieter Lang, Jun 22 2011

The reduced residue system modulo n used here is the set of numbers k from the set {0,1,...,n-1} which satisfy gcd(k,n)=1. There are phi(n) = A000010(n) such numbers k.
This is the m=4 member of a family of sequences, call them rmnS(m) (reduced mod n sum), with entries rmnS(m;n):=sum(binomial(k+m-1,m),0<=k<=n-1 with gcd(k,n)=1), m>=0, n>=1. Recall gcd(0,n)=n.
The members for m=0, 1, 2 and 3 are A000010, A023896, A127415, and A189918, respectively, where in the m=1 and 2 cases the offset for n=1 should be taken as 0 (not 1).

			a(6) = A000332(4) + A000292(8)= 1 + 70 = 71.
a(6) = (6/6!)*(6*3666*(1/3) + 5*137*2 - 182) = 71.
a(12) = A000332(4) + A000332(8) + A000332(10) + A000332(14) = 1 + 70 + 210 + 1001 = 1282.
a(12) = (12/6!)*(12*18258*(1/3) + 5*407*2 - 182) = 1282.

Cf. A189918, A023896, A127415, A189922.

a(n) = sum(k=0, n-1, if (gcd(n,k) == 1, binomial(k+3, 4))); \\ Michel Marcus, Feb 01 2016

a(n) = sum(A000332(k+3), 0<=k<=n-1, gcd(k,n)=1), n>=1.

a(n) = (n/6!)*(n*(6*n^3+45*n^2+110*n+90)*P(1,n) + 5*(2*n^2+9*n+11)*P(-1,n) - P(-3,n)), n>=2, with P(k,n):= J(k,n)/n^k, where J(k,n) is the Jordan function (see A000010, A007434, A059376 - A059378, A069091 - A069095).

More terms from Michel Marcus, Feb 01 2016

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A046970 Dirichlet inverse of the Jordan function J_2 (A007434).

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

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A321222 a(n) = Sum_{d|n} mu(d)*d^n.

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A334660 Dirichlet g.f.: 1 / zeta(s-4).

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A189923 Jordan function J_{-5}(n) multiplied by n^5.

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A322324 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Product_{p|n, p prime} (1 - p^k).

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A192000 Sum of binomial numbers A000332(k+3), with k in the reduced residue system modulo n.