A063453 -id:A063453 - cp's OEIS frontend

A059376 Jordan function J_3(n).

1, 7, 26, 56, 124, 182, 342, 448, 702, 868, 1330, 1456, 2196, 2394, 3224, 3584, 4912, 4914, 6858, 6944, 8892, 9310, 12166, 11648, 15500, 15372, 18954, 19152, 24388, 22568, 29790, 28672, 34580, 34384, 42408, 39312, 50652, 48006, 57096

Offset: 1

N. J. A. Sloane, Jan 28 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

T. D. Noe, Table of n, a(n) for n = 1..1000
D. H. Lehmer, On a theorem of von Sterneck, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)
Wikipedia, Jordan's totient function.

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059377 (J_4), A059378 (J_5), A069091 - A069095 (J_6 through J_10).

Cf. A013662, A215267.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 3)
A059376 := proc(n)
    add(d^3*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Nov 03 2015

JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 3]; Array[f, 39]
f[p_, e_] := p^(3*e) - p^(3*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,120,print1(sumdiv(n,d,d^3*moebius(n/d)),","))

for (n = 1, 1000, write("b059376.txt", n, " ", sumdiv(n, d, d^3*moebius(n/d))); ) \\ Harry J. Smith, Jun 26 2009

seq(n) = dirmul(vector(n,k,k^3), vector(n,k,moebius(k)));
seq(39)  \\ Gheorghe Coserea, May 11 2016

from math import prod
from sympy import factorint
def A059376(n): return prod(p**(3*(e-1))*(p**3-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

Multiplicative with a(p^e) = p^(3e) - p^(3e-3). - Vladeta Jovovic, Jul 26 2001
a(n) = Sum_{d|n} d^3*mu(n/d). - Benoit Cloitre, Apr 05 2002
Dirichlet generating function: zeta(s-3)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
A063453(n) divides a(n). - R. J. Mathar, Mar 30 2011
a(n) = Sum_{k=1..n} gcd(k,n)^3 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
a(n) = n^3*Product_{distinct primes p dividing n} (1-1/p^3). - Tom Edgar, Jan 09 2015
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 4*x + x^2)/(1 - x)^4. - Ilya Gutkovskiy, Apr 25 2017
Sum_{d|n} a(d) = n^3. - Werner Schulte, Jan 12 2018
Sum_{k=1..n} a(k) ~ 45*n^4 / (2*Pi^4). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^3 = 1/zeta(4) (A215267).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^3/(p^3-1)^2) = 1.2253556451... (End)
O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 4*x^n + x^(2*n))/(1 - x^n)^4 = x + 7*x^2 + 26*x^3 + 56*x^4 + 124*x^5 + .... - Peter Bala, Jan 31 2022
From Peter Bala, Jan 01 2024: (Start)
a(n) = Sum_{d divides n} d * J_2(d) * phi(n/d) = Sum_{d divides n} d^2 * phi(d) * J_2(n/d), where J_2(n) = A007434(n).
a(n) = Sum_{k = 1..n} gcd(k, n) * J_2(gcd(k, n)) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_1(gcd(j, k, n)). (End)
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i)*J_2(j) = Sum_{1 <= i, j, k <= n, lcm(i, j, k) = n} phi(i)*phi(j)*phi(k), where J_2(n) = A007434(n). - Peter Bala, Jan 29 2024

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

Original entry on oeis.org

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

A053820 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^4.

Original entry on oeis.org

1, 1, 17, 82, 354, 626, 2275, 3108, 7395, 9044, 25333, 17668, 60710, 50470, 88388, 103496, 243848, 129750, 432345, 266088, 497574, 497178, 1151403, 539912, 1541770, 1153724, 1900089, 1516844, 3756718, 1246568, 5273999

Offset: 1

N. J. A. Sloane, Apr 07 2000

nonn

If gcd(n,30) = 1, then a(n) is divisible by n. If n has at least one prime factor == 1 (mod 30), then a(n) is divisible by n. - Jianing Song, Jul 13 2018

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_4(n).
L. E. Dickson, History of the Theory of Numbers, Vol. I (Reprint 1966), p. 140.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (2005) 403-408.

Column k=4 of A308477.

Cf. A000010, A023900, A053818, A053819, A063453.

a[n_] := Sum[If[GCD[n, k] == 1, k^4, 0], {k, 1, n}]; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Feb 26 2014 *)
a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^4/5) * Times @@ ((p - 1)*p^(e - 1)) + (n^3/3) * Times @@ (1 - p) - (n/30) * Times @@ (1 - p^3)]; Array[a, 100] (* Amiram Eldar, Dec 03 2023 *)

a(n) = sum(k=1, n, (gcd(n,k) == 1)*k^4); \\ Michel Marcus, Feb 26 2014

a(n) = {my(f = factor(n)); if(n == 1, 1, (n^4/5) * eulerphi(f) + (n^3/3) * prod(i = 1, #f~, 1 - f[i, 1]) - (n/30) * prod(i = 1, #f~, 1 - f[i, 1]^3));} \\ Amiram Eldar, Dec 03 2023

a(n) = (6*n^4*A000010(n)+10*n^3*A023900(n)-n*A063453(n))/30 for n>1. Formula is derived from a more general formula of A. Thacker (1850), see [Dickson, Brown]. - Franz Vrabec, Aug 21 2005
G.f. A(x) satisfies: A(x) = x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^6 - Sum_{k>=2} k^4 * A(x^k). - Ilya Gutkovskiy, Mar 29 2020
Sum_{k=1..n} a(k) ~ n^6 / (5*Pi^2). - Amiram Eldar, Dec 03 2023

A011785 Number of 3 X 3 matrices whose determinant is 1 mod n.

Original entry on oeis.org

1, 168, 5616, 43008, 372000, 943488, 5630688, 11010048, 36846576, 62496000, 212427600, 241532928, 810534816, 945955584, 2089152000, 2818572288, 6950204928, 6190224768, 16934047920, 15998976000, 31621943808, 35687836800

Offset: 1

Benjamin T. Love (benlove(AT)preston.polaristel.net)

Order of the group SL(3,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Nov 24 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Cf. A000010, A000578, A003557, A003800, A046970, A063453.
Cf. A000056 (SL(2,Z_n)), A011786 (SL(4,Z_n)).
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).

a[n_] := (n^9*Times @@ Function[p, (1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)] /@ FactorInteger[n][[All, 1]])/EulerPhi[n]; a[1] = 1; Array[a, 30] (* Jean-François Alcover, Mar 21 2017 *)

a(n) = n^9*prod(k=2, n, if (!isprime(k) || (n % k), 1, (1-1/k^3)*(1-1/k^2)*(1-1/k)))/eulerphi(n); \\ Michel Marcus, Jun 30 2015

from math import prod
from sympy import factorint
def A011785(n): return prod(p**((e<<3)-5)*(p**2*(p*(p-1)*(p+1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = p^(8*e-5)*(p^3 - 1)*(p^2 - 1). - Vladeta Jovovic, Nov 18 2001
For a formula see A064767.
a(n) = A046970(n)*A063453(n)*A000578(n)*A003557(n)^5. - R. J. Mathar, Mar 30 2011
a(n) = A064767(n)/phi(n). - Jianing Song, Nov 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/((p-1)^3 * (p+1)^2 * (p^2 + p + 1) * (p^6 + p^4 + p^2 + 1))) = 1.0061577672748872278355775942508642214184417621389767880397578015151659965... - Vaclav Kotesovec, Sep 19 2020
Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 - (p^3 + p^2 -1)/p^6) = 0.08630488937... . - Amiram Eldar, Oct 23 2022

More terms from John W. Layman, Feb 16 2001

Further terms from Vladeta Jovovic, Oct 29 2001

A189922 Jordan function J_{-4} multiplied by n^4.

Original entry on oeis.org

1, -15, -80, -15, -624, 1200, -2400, -15, -80, 9360, -14640, 1200, -28560, 36000, 49920, -15, -83520, 1200, -130320, 9360, 192000, 219600, -279840, 1200, -624, 428400, -80, 36000, -707280, -748800, -923520, -15, 1171200, 1252800, 1497600, 1200, -1874160

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 - 2^4 = -15.
a(4) = mu(1)*1^4 + mu(2)*2^4 + mu(4)*4^4 = 1 - 16 + 0 = -15.
Sum identity for n=4: a(1)*(4/1)^4 + a(2)*(4/2)^4 + a(4)*(4/4)^4 = 256 - 15*16 - 15 = 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 (k=-1), A046970 (k=-2), A063453 (k=-3).

a:= n-> mul(1-i[1]^4, i=ifactors(n)[2]):
seq(a(n), n=1..48);  # Alois P. Heinz, Jan 26 2024

a[n_] := Sum[ MoebiusMu[d]*d^4, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := (1-p^4); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)

for (n=1, 30, print1(sumdiv(n, d, moebius(d) * d^4),", ")); \\ Indranil Ghosh, Mar 11 2017

a(n) = J_{-4}(n)*n^4 = Product_{p prime | n} (1 - p^4), for n>=2, a(1)=1.
a(n) = Sum_{d|n} mu(d)*d^4 with the Moebius function mu = A008683.
Dirichlet g.f.: zeta(s)/zeta(s-4).
Sum identity: Sum_{d|n} a(n)*(n/d)^4 = 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^4*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
a(n) = Sum_{d divides n} d * sigma_3(d)^(-1) * sigma_1(n/d), where sigma_3(n)^(-1) = A053825(n) denotes the Dirichlet inverse of sigma_3(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).

A334659 Dirichlet g.f.: 1 / zeta(s-3).

Original entry on oeis.org

1, -8, -27, 0, -125, 216, -343, 0, 0, 1000, -1331, 0, -2197, 2744, 3375, 0, -4913, 0, -6859, 0, 9261, 10648, -12167, 0, 0, 17576, 0, 0, -24389, -27000, -29791, 0, 35937, 39304, 42875, 0, -50653, 54872, 59319, 0, -68921, -74088, -79507, 0, 0, 97336, -103823, 0, 0, 0, 132651, 0, -148877

Offset: 1

Ilya Gutkovskiy, May 07 2020

Dirichlet inverse of A000578.
Moebius transform of A063453.
Inverse Moebius transform of A053825.

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

Cf. A008683, A055615, A334657, A334660.

Cf. A000578, A053825, A063453.

Mathematica
```
Table[MoebiusMu[n] n^3, {n, 53}]
```

G.f. A(x) satisfies: A(x) = x - 2^3 * A(x^2) - 3^3 * A(x^3) - 4^3 * A(x^4) - ...
a(1) = 1; a(n) = -n^3 * Sum_{d|n, d < n} a(d) / d^3.
a(n) = mu(n) * n^3.
Multiplicative with a(p^e) = -p^3 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

A328640 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-3)).

Original entry on oeis.org

1, -9, -28, 9, -126, 252, -344, -9, 28, 1134, -1332, -252, -2198, 3096, 3528, 9, -4914, -252, -6860, -1134, 9632, 11988, -12168, 252, 126, 19782, -28, -3096, -24390, -31752, -29792, -9, 37296, 44226, 43344, 252, -50654, 61740, 61544, 1134, -68922, -86688, -79508, -11988, -3528

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A065959.

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

Cf. A008683, A008836, A026424 (positions of negative terms), A063453, A065959, A323363, A328639.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^3, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^3 &], {n, 1, 45}]
f[p_, e_] := (-1)^e*(p^3+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^3)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA065959(n/d) * a(d).
a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^3, where lambda = A008836 and mu = A008683.
Multiplicative with a(p^) = (-1)^e*(p^3+1). - Amiram Eldar, Dec 03 2022

A069056 Numbers k such that Sum_{d|k} d^2*mu(d) divides k^2.

Original entry on oeis.org

1, 12, 24, 36, 48, 72, 96, 108, 120, 144, 192, 216, 240, 288, 324, 336, 360, 384, 432, 480, 576, 600, 648, 672, 720, 768, 864, 960, 972, 1008, 1080, 1152, 1200, 1296, 1344, 1440, 1536, 1728, 1800, 1920, 1944, 2016, 2160, 2304, 2352, 2400, 2448, 2592, 2688

Offset: 1

Benoit Cloitre, Apr 07 2002

Numbers k such that A046970(k) divides k^2. [corrected by Amiram Eldar, Apr 20 2025]

If n > 1, a(n+1) - a(n) == 0 (mod 12), so a(n+1) - a(n) = 12 for n=2,3,4,5,7,8,...; a(n+1) - a(n) = 24 for n=6,9,.... Conjecture: if c > 2 and n > 1, Sum_{d|n} d^c*mu(d) never divides n^c. Hence A063453(n) never divides n^3 for n > 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A046970, A063453.

a069056 n = a069056_list !! (n-1)
a069056_list = filter (\x -> x ^ 2 `mod` a046970 x == 0) [1..]
-- Reinhard Zumkeller, Jan 19 2012

f[d_] := d^2*MoebiusMu[d]; ok[n_] := Divisible[n^2, Total[f /@ Divisors[n]]]; Select[Range[3000], ok] (* Jean-François Alcover, Nov 15 2011 *)
q[k_] := Divisible[k^2, Times @@ ((First[#]^2-1)& /@ FactorInteger[k])]; q[1] = True; Select[Range[3000], q] (* Amiram Eldar, Apr 20 2025 *)

for(n=1,1000,if(n^2%sumdiv(n,d,moebius(d)*d^2)==0,print1(n,",")))

isok(k) = {my(f = factor(k)); !(k^2 % prod(i = 1, #f~, f[i,1]^2-1));} \\ Amiram Eldar, Apr 20 2025

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