A059376 -id:A059376 - cp's OEIS frontend

A255807 E.g.f.: exp(Sum_{k>=1} k^2 * x^k).

1, 1, 9, 79, 841, 10821, 162601, 2777419, 52960209, 1112813641, 25509407401, 632772511911, 16870674740569, 480717000225229, 14568646143888201, 467640968478534691, 15841420612530533281, 564519727866573515409, 21102817266052772063689, 825435163723385398719871

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if e.g.f. = exp(Sum_{k>=1} k^m * x^k) and m>0, then a(n) ~ (m+2)^(-1/2) * Gamma(m+2)^(1/(2*m+4)) * exp((m+2)/(m+1) * Gamma(m+2)^(1/(m+2)) * n^((m+1)/(m+2)) + zeta(-m) - n) * n^(n - 1/(2*m+4)).
It appears that the sequence a(n) taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k: if true, then the sequence a(n) taken modulo k would be periodic with period dividing k. - Peter Bala, Nov 14 2017
The above conjecture is true - see the Bala link. - Peter Bala, Jan 20 2018

G. C. Greubel, Table of n, a(n) for n = 0..250
P. Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A082579, A255819, A000262.

nmax=20; CoefficientList[Series[Exp[Sum[k^2*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k^2*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)

E.g.f.: exp(x*(1+x)/(1-x)^3).
a(n) ~ 2^(-7/8) * 3^(1/8) * n^(n-1/8) * exp(2^(9/4) * 3^(-3/4) * n^(3/4) - n).
a(n) = n!*y(n) where y(0)=1 and y(n)=(Sum_{k=0..n-1} (n-k)^3*y(k))/n for n>=1. - Benedict W. J. Irwin, Jun 02 2016
a(n) = (4*n-3)*a(n-1) - 2*(n-1)*(3*n-8)*a(n-2) + (n-1)*(n-2)*(4*n-11)*a(n-3) - (n-1)*(n-2)*(n-3)*(n-4)*a(n-4). - Peter Bala, Nov 12 2017
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_3(k)/k), where J_3() is the Jordan function (A059376). - Ilya Gutkovskiy, May 25 2019

A059382 Product J_3(i), i=1..n.

Original entry on oeis.org

1, 7, 182, 10192, 1263808, 230013056, 78664465152, 35241680388096, 24739659632443392, 21474024560960864256, 28560452666077949460480, 41584019081809494414458880, 91318505903653649734151700480, 218616503133346837463559170949120

Offset: 1

N. J. A. Sloane, Jan 28 2001

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j)^3 for 1 <= i,j <= n. - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 203, #17.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..100
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Eric Weisstein's World of Mathematics, Le Paige's Theorem

Cf. A001088, A059376, A059381, A059383, A059384, A175836.

JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n]; A059382[n_]:=Times@@(JordanTotient[#, 3]&/@Range[n]); (* Enrique Pérez Herrero, Aug 06 2011 *)

A069092 Jordan function J_7(n).

Original entry on oeis.org

1, 127, 2186, 16256, 78124, 277622, 823542, 2080768, 4780782, 9921748, 19487170, 35535616, 62748516, 104589834, 170779064, 266338304, 410338672, 607159314, 893871738, 1269983744, 1800262812, 2474870590, 3404825446, 4548558848

Offset: 1

Benoit Cloitre, Apr 05 2002

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A069091 (J_6), A069092 (J_7), A069093 (J_8), A069094 (J_9), A069095 (J_10). [Enrique Pérez Herrero, Nov 02 2010]
Cf. A013666.

JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
A069092[n_] := JordanTotient[n, 7]; (* Enrique Pérez Herrero, Nov 02 2010 *)
f[p_, e_] := p^(7*e) - p^(7*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1, 100, print1(sumdiv(n, d, d^7*moebius(n/d)), ", "))

a(n) = Sum_{d|n} d^7*mu(n/d).
Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).
Dirichlet generating function: zeta(s-7)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A_7(x^n)/(1 - x^n)^8 = x + 127*x^2 + 2186*x^3 + 16256*x^4 + 78124*x^5 + ..., where A_7(x) = x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7 is the 7th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A069094 Jordan function J_9(n).

Original entry on oeis.org

1, 511, 19682, 261632, 1953124, 10057502, 40353606, 133955584, 387400806, 998046364, 2357947690, 5149441024, 10604499372, 20620692666, 38441386568, 68585259008, 118587876496, 197961811866, 322687697778, 510999738368

Offset: 1

Benoit Cloitre, Apr 05 2002

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).

Cf. A013668.

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 9]; Array[f, 22]
f[p_, e_] := p^(9*e) - p^(9*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(sumdiv(n,d,d^9*moebius(n/d)),","))

a(n) = Sum_{d|n} d^9*mu(n/d).
Multiplicative with a(p^e) = p^(9e)-p^(9(e-1)).
Dirichlet generating function: zeta(s-9)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^9*Product_{distinct primes p dividing n} (1-1/p^9). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 18711*n^10 / (2*Pi^10). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^9 = 1/zeta(10).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^9/(p^9-1)^2) = 1.0020122252...  (End)

A368743 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^3.

Original entry on oeis.org

1, 11, 35, 100, 149, 385, 391, 848, 1017, 1639, 1451, 3500, 2365, 4301, 5215, 6976, 5201, 11187, 7219, 14900, 13685, 15961, 12695, 29680, 19225, 26015, 28107, 39100, 25229, 57365, 30751, 56576, 50785, 57211, 58259, 101700, 52021, 79409, 82775, 126352

Offset: 1

Peter Bala, Jan 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A007434, A059376, A069097, A343497, A343498, A360428.

seq( add(add(igcd(i, j, n)^3, i = 1..n), j = 1..n), n = 1..50);
# faster program for large n
with(numtheory):
A007434 := proc(n) add(d^2*mobius(n/d), d in divisors(n)) end proc:
seq( add(d^3*A007434(n/d), d in divisors(n)), n = 1..500);

f[p_, e_] := p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 29 2024 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1));} \\ Amiram Eldar, Jan 29 2024

from math import prod
from sympy import factorint
def A368743(n): return prod(p**(e-1<<1)*(p**e*(p*(q:=p+1)+1)-q) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n)^2.
a(n) = Sum_{d divides n} d^3 * J_2(n/d) = Sum_{d divides n} d^2 * J_3(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n).
Dirichlet g.f.: zeta(s-2) * zeta(s-3)/zeta(s).
a(n) is a multiplicative function: for prime p, a(p^k) = p^(3*k-2)*(p^2 + p + 1) - p^(2*k-2)*(p + 1).
Sum_{k=1..n} a(k) ~ c * n^4, where c = 15/(4*Pi^2) = 0.379954... . - Amiram Eldar, Jan 29 2024
a(n) = Sum_{d divides n} mobius(n/d) * d^2 * sigma(d). - Peter Bala, Jan 29 2024

A263825 Total number c_{pi_1(B_1)}(n) of n-coverings over the first amphicosm.

Original entry on oeis.org

1, 7, 5, 23, 7, 39, 9, 65, 18, 61, 13, 143, 15, 87, 35, 183, 19, 182, 21, 245, 45, 151, 25, 465, 38, 189, 58, 375, 31, 429, 33, 549, 65, 277, 63, 806, 39, 327, 75, 875, 43, 663, 45, 719, 126, 439, 49, 1535, 66, 650, 95, 933, 55, 982, 91, 1425, 105, 637, 61, 2093

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..20000
G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
G. Chelnokov, M. Deryagina and A. Mednykh, On the coverings of Euclidean manifolds B_1 and B_2, Communications in Algebra, Vol. 45, No. 4 (2017), 1558-1576.

Cf. A263825-A263830, A263832.

A263825 := proc(n)
    local a,l,m,s1,s2,s3,s4 ;
    # Theorem 2
    a := 0 ;
    for l in numtheory[divisors](n) do
        m := n/l ;
        s1 := 0 ;
        for twok in numtheory[divisors](m) do
            if type(twok,'even') then
                k := twok/2 ;
                s1 := s1+numtheory[sigma](k)*k ;
            end if;
        end do:
        s2 := 0 ;
        for d in numtheory[divisors](l) do
            s2 := s2+numtheory[mobius](l/d)*d^2*igcd(2,d) ;
        end do:
        s3 := 0 ;
        for k in numtheory[divisors](m) do
            s3 := s3+numtheory[sigma](m/k)*k ;
            if modp(m,2*k) = 0 then
                s3 := s3-numtheory[sigma](m/2/k)*k ;
            end if;
        end do:
        s4 := 0 ;
        for twok in numtheory[divisors](m) do
            if type(twok,'even') then
                s4 := s4+numtheory[sigma](m/twok)*twok ;
                if modp(m,2*twok) = 0 then
                    s4 := s4-numtheory[sigma](m/2/twok)*twok ;
                end if;
            end if;
        end do:
        a := a+A059376(l)*s1 + s2*s3 + A007434(l)*s4 ;
    end do:
    a/n ;
end proc: # R. J. Mathar, Nov 03 2015

A007434[n_] := Sum[ MoebiusMu[n/d] * d^2, {d, Divisors[n]}];
A059376[n_] := Sum[ MoebiusMu[n/d] * d^3, {d, Divisors[n]}];
A263825[n_] := Module[{a, l, m, s1, s2, s3, s4},
a = 0;
Do[m = n/l;
s1 = 0; Do[If[EvenQ[twok], k = twok/2; s1 = s1 + DivisorSigma[1, k]*k], {twok, Divisors[m]}];
s2 = 0; Do[s2 = s2 + MoebiusMu[l/d]*d^2*GCD[2, d], {d, Divisors[l]}];
s3 = 0; Do[s3 = s3 + DivisorSigma[1, m/k]*k ; If[Mod[m, 2*k] == 0, s3 = s3 - DivisorSigma[1, m/2/k]*k], {k, Divisors[m]}];
s4 = 0; Do[If[EvenQ[twok], s4 = s4 + DivisorSigma[1, m/twok]*twok; If[ Mod[m, 2*twok] == 0, s4 = s4 - DivisorSigma[1, m/2/twok]*twok]], {twok, Divisors[m]}]; a = a + A059376[l]*s1 + s2*s3 + A007434[l]*s4,
{l, Divisors[n]}]; a/n
];
Array[A263825, 60] (* Jean-François Alcover, Nov 21 2017, after R. J. Mathar *)

A001001(n) = sumdiv(n, d, sigma(d) * d);
A007434(n) = sumdiv(n, d, moebius(n\d) * d^2);
A059376(n) = sumdiv(n, d, moebius(n\d) * d^3);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
EpiPcZn(n) = sumdiv(n, d, moebius(n\d) * d^2 * gcd(d,2));
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
a(n) = { 1/n * sumdiv(n, d,
  A059376(d) * S1(n\d) + EpiPcZn(d) * S11(n\d) + A007434(d) * S21(n\d));
};
vector(60, n, a(n))  \\ Gheorghe Coserea, May 04 2016

A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

Original entry on oeis.org

0, 1, 3, 9, 12, 25, 27, 49, 48, 81, 75, 121, 108, 169, 147, 225, 192, 289, 243, 361, 300, 441, 363, 529, 432, 625, 507, 729, 588, 841, 675, 961, 768, 1089, 867, 1225, 972, 1369, 1083, 1521, 1200, 1681, 1323, 1849, 1452, 2025, 1587, 2209, 1728, 2401, 1875, 2601, 2028, 2809, 2187, 3025

Offset: 0

Ilya Gutkovskiy, May 26 2019

Moebius transform of A076577.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A007434, A016754, A026741, A033428, A076577, A308418, A309337.

a[n_] := If[OddQ[n], n^2, 3 n^2/4]; Table[a[n], {n, 0, 55}]
nmax = 55; CoefficientList[Series[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3, {x, 0, nmax}], x]
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 3, 9, 12, 25}, 56]
Table[(7 - (-1)^n) n^2/8, {n, 0, 55}]

G.f.: x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3.
G.f.: Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k)), where J_2() is the Jordan function (A007434).
E.g.f.: x*((4 + 3*x)*cosh(x) + (3 + 4*x)*sinh(x))/4.
Dirichlet g.f.: zeta(s-2)*(1 - 1/2^s).
a(n) = (7 - (-1)^n)*n^2/8.
a(n) = Sum_{d|n, n/d odd} J_2(d).
a(2*k+1) = A016754(k), a(2*k) = A033428(k).
Sum_{n>=1} 1/a(n) = 13*Pi^2/72 = 1.7820119057522453061...
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/72 = 0.68538919452009434853...
Multiplicative with a(2^e) = 3*2^(2*e-2), and a(p^e) = p^(2*e) for odd primes p. - Amiram Eldar, Oct 26 2020
For n >= 1, n*a(n) = A309337(n) = Sum_{d divides n} (-1)^(d+1) * J(3, n/d), where the Jordan totient function J_3(n) = A059376. - Peter Bala, Jan 21 2024

A309337 a(n) = n^3 if n odd, 3*n^3/4 if n even.

Original entry on oeis.org

0, 1, 6, 27, 48, 125, 162, 343, 384, 729, 750, 1331, 1296, 2197, 2058, 3375, 3072, 4913, 4374, 6859, 6000, 9261, 7986, 12167, 10368, 15625, 13182, 19683, 16464, 24389, 20250, 29791, 24576, 35937, 29478, 42875, 34992, 50653, 41154, 59319, 48000, 68921, 55566, 79507, 63888, 91125

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A078307.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A000578, A016755, A059376, A078307, A129194, A193356, A309338.

a[n_] := If[OddQ[n], n^3, 3 n^3/4]; Table[a[n], {n, 0, 45}]
nmax = 45; CoefficientList[Series[x (1 + 6 x + 23 x^2 + 24 x^3 + 23 x^4 + 6 x^5 + x^6)/(1 - x^2)^4, {x, 0, nmax}], x]
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 6, 27, 48, 125, 162, 343}, 46]
Table[n^3 (7 - (-1)^n)/8, {n, 0, 45}]

G.f.: x * (1 + 6*x + 23*x^2 + 24*x^3 + 23*x^4 + 6*x^5 + x^6)/(1 - x^2)^4.
G.f.: Sum_{k>=1} J_3(k) * x^k/(1 + x^k), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) * (1 - 2^(1-s)).
a(n) = n^3 * (7 - (-1)^n)/8.
a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_3(d).
Sum_{n>=1} 1/a(n) = 25*zeta(3)/24 = 1.252142607457910713958...
Multiplicative with a(2^e) = 3*2^(3*e-2), and a(p^e) = p^(3*e) for odd primes p. - Amiram Eldar, Oct 26 2020
a(n) = Sum_{1 <= i, j, k <= n} (-1)^(1 + gcd(i,j,k,n)) = Sum_{d | n} (-1)^(d+1) * J_3(n/d). Cf. A129194. - Peter Bala, Jan 16 2024

A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

Original entry on oeis.org

1, 106, 1041, 7218, 19345, 110346, 136801, 465522, 768327, 2050570, 1947121, 7513938, 5226481, 14500906, 20138145, 29822066, 25640641, 81442662, 49651921, 139632210, 142409841, 206394826, 154751521, 484608402, 302749845, 554006986, 560366223, 987429618, 616040881

Offset: 1

Seiichi Manyama, May 26 2024

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

Cf. A059376, A084220.
Cf. A373130, A373131.
Cf. A062952, A373132, A373135.
Cf. A013662, A013665.

f[p_, e_] := (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=3) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{d|n} J_3(d) * sigma(d^3), where the Jordan totient function J_3(n) = A059376(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(4) * zeta(7) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 + 1/p^8) = 1.71945569563704656468... . (End)

A115224 Number of 3 X 3 symmetric matrices over Z(n) having determinant 1.

Original entry on oeis.org

1, 28, 234, 896, 3100, 6552, 16758, 28672, 56862, 86800, 160930, 209664, 371124, 469224, 725400, 917504, 1419568, 1592136, 2475738, 2777600, 3921372, 4506040, 6435814, 6709248, 9687500, 10391472, 13817466, 15015168, 20510308, 20311200, 28628190, 29360128, 37657620

Offset: 1

T. D. Noe, Jan 16 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4885 from Enrique Pérez Herrero)

Cf. A000056 (order of the group SL(2, Z_n)), A011785 (number of 3 X 3 matrices whose determinant is 1 mod n, i.e. order of SL(3, Z_n)).

Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]==1, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}]; cnt, {n, 2, 20}]
JordanTotient[n_,k_:1] := DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]; A115224[n_IntegerQ] := JordanTotient[n^2,3]/n; Table[A115224[n], {n,100}] (* Enrique Pérez Herrero, Sep 14 2010 *)
f[p_, e_] := (p^3 - 1)*p^(5*e - 3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 15 2020 *)

a(1)=1 because the matrix of all zeros has determinant 0, but 0=1 (mod 1).
For prime p, a(p) = (p^3-1)*p^2.
Multiplicative with a(p^e) = (p^3-1)*p^(5e-3).
a(n) = A011785(n)/A000056(n).
a(n) = A059376(n^2)/n. - Enrique Pérez Herrero, Sep 14 2010
a(n) = n^2*A059376(n). Dirichlet g.f.: zeta(s-5)/zeta(s-2). - R. J. Mathar, Feb 27 2011
Sum_{k=1..n} a(k) ~ 15*n^6 / Pi^4. - Vaclav Kotesovec, Feb 07 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/(1 - p^3 - p^5 + p^8)) = 1.04172462829914219180789244796430293454403616906393417764614215669994022537... - Vaclav Kotesovec, Sep 20 2020

cp's OEIS Frontend

A255807 E.g.f.: exp(Sum_{k>=1} k^2 * x^k).

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A059382 Product J_3(i), i=1..n.

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A069092 Jordan function J_7(n).

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A069094 Jordan function J_9(n).

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A368743 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^3.

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A263825 Total number c_{pi_1(B_1)}(n) of n-coverings over the first amphicosm.

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A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

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A309337 a(n) = n^3 if n odd, 3*n^3/4 if n even.

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