A013663 -id:A013663 - cp's OEIS frontend

A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

1, 13, 73, 205, 601, 949, 2353, 3277, 5905, 7813, 14521, 14965, 28393, 30589, 43873, 52429, 83233, 76765, 129961, 123205, 171769, 188773, 279313, 239221, 375601, 369109, 478297, 482365, 706441, 570349, 922561, 838861, 1060033, 1082029

Offset: 1

Benoit Cloitre, Jun 21 2003

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001157, A001159, A002117, A007434, A013663, A065827.
Cf. A068963, A372962, A372963, A373007.
Cf. A057660, A084220, A372966, A373105.

with(numtheory): a:=n->sigma[4](n^2)/sigma[2](n^2): seq(a(n),n=1..40); # Muniru A Asiru, Oct 09 2018

Table[DivisorSigma[4, n^2]/DivisorSigma[2, n^2], {n, 1, 50}] (* G. C. Greubel, Oct 08 2018 *)
f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)

a(n)=sumdiv(n^2,d,d^4)/sumdiv(n^2,d,d^2)

a(n) = sigma(n^2, 4)/sigma(n^2, 2); \\ Michel Marcus, Oct 09 2018

Multiplicative with a(p^e) = (p^(4*e + 2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.09957644430375183822287768590764825667080036406680891521221069625517483696... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(3)) = 0.172525... . - Amiram Eldar, Oct 30 2022
From Peter Bala, Jan 18 2024: (Start)
a(n) = Sum_{d divides n} J_2(d^2) = Sum_{d divides n} d^2 * J_2(d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{1 <= j, k <= n} ( n/gcd(j, k, n) )^2.
Dirichlet g.f.: zeta(s) * zeta(s-4) / zeta(s-2) [Corrected by Michael Shamos, May 18 2025]. (End)
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_4(d). - Seiichi Manyama, May 18 2024

A244667 Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^2) where H(n) is the n-th harmonic number.

Original entry on oeis.org

9, 7, 5, 4, 2, 6, 2, 5, 1, 3, 8, 7, 2, 5, 7, 0, 5, 6, 5, 6, 8, 2, 3, 2, 6, 5, 8, 9, 9, 1, 2, 8, 8, 1, 8, 3, 2, 5, 1, 0, 2, 5, 8, 3, 6, 2, 9, 2, 4, 4, 8, 0, 2, 9, 8, 5, 0, 2, 2, 6, 7, 3, 6, 1, 3, 3, 3, 2, 4, 1, 9, 5, 7, 5, 4, 3, 7, 1, 5, 3, 4, 1, 9, 0, 2, 7, 0, 7, 6, 7, 1, 7, 0, 0, 2, 4, 9, 6, 3, 0, 2

Offset: 1

Jean-François Alcover, Jul 04 2014

			9.75426251387257056568232658991288183251025836292448029850226736133324...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 27.

Cf. A001008, A002805, A002117, A013663.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); Pi(R)^2/6*Evaluate(L,3) + 15/2*Evaluate(L,5); // G. C. Greubel, Aug 31 2018

RealDigits[15/2*Zeta[5] + Zeta[2]*Zeta[3], 10, 101] // First

default(realprecision, 100); Pi^2/6*zeta(3) + 15/2*zeta(5) \\ G. C. Greubel, Aug 31 2018

Equals Pi^2/6*zeta(3) + 15/2*zeta(5).

A008382 a(n) = floor(n/5)floor((n+1)/5)floor((n+2)/5)floor((n+3)/5)floor((n+4)/5).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 48, 72, 108, 162, 243, 324, 432, 576, 768, 1024, 1280, 1600, 2000, 2500, 3125, 3750, 4500, 5400, 6480, 7776, 9072, 10584, 12348, 14406, 16807, 19208, 21952, 25088, 28672, 32768, 36864, 41472, 46656, 52488, 59049, 65610, 72900, 81000

Offset: 0

N. J. A. Sloane

nonn

For n >= 5, a(n) is the maximal product of 5 positive integers with sum n. - Wesley Ivan Hurt, Jun 29 2022

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,4,-8,4,0,0,-6,12,-6,0,0,4,-8,4,0,0,-1,2,-1).

Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), this sequence (k=5), A008881 (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

Cf. A000584, A013663.

CoefficientList[Series[x^5*(x^10 - 2*x^9 + 4*x^8 - 4*x^7 + 8*x^6 - 8*x^5 + 8*x^4 - 4*x^3 + 4*x^2 - 2*x + 1)*(1 + x)^2/((x^4 + x^3 + x^2 + x + 1)^4*(x - 1)^6), {x, 0, 60}], x] (* Wesley Ivan Hurt, Jun 29 2022 *)

A008382(n):=floor(n/5)*floor((n+1)/5)*floor((n+2)/5)*floor((n+3)/5)*floor((n+4)/5)$
makelist(A008382(n),n,0,30); /* Martin Ettl, Oct 26 2012 */

From R. J. Mathar, May 08 2013: (Start)
a(n) = +2*a(n-1) -a(n-2) +4*a(n-5) -8*a(n-6) +4*a(n-7) -6*a(n-10) +12*a(n-11) -6*a(n-12) +4*a(n-15) -8*a(n-16) +4*a(n-17) -a(n-20) +2*a(n-21) -a(n-22).
G.f.: x^5 *(x^10 -2*x^9 +4*x^8 -4*x^7 +8*x^6 -8*x^5 +8*x^4 -4*x^3 +4*x^2 -2*x+1) *(1+x)^2 / ( (x^4+x^3+x^2+x+1)^4 *(x-1)^6 ). (End)
a(5*m) = m^5 (A000584). - Bernard Schott, Sep 21 2022
Sum_{n>=5} 1/a(n) = 1 + zeta(5). - Amiram Eldar, Jan 10 2023

A206623 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^3).

Original entry on oeis.org

1, 2, 18, 88, 398, 1768, 7508, 30644, 121310, 467234, 1756080, 6457168, 23274788, 82381584, 286760344, 982874120, 3320800590, 11070619228, 36446345198, 118581503192, 381552358872, 1214868568728, 3829841265428, 11959828895612, 37013411304892, 113570015855642

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Convolution of A023872 and A248882. - Vaclav Kotesovec, Aug 19 2015

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^8/(1-x^2)^8 * (1+x^3)^27/(1-x^3)^27 *...
Also, A(x) = Euler transform of [2,15,54,120,250,405,686,960,1458,...]:
A(x) = 1/((1-x)^2*(1-x^2)^15*(1-x^3)^54*(1-x^4)^120*(1-x^5)^250*(1-x^6)^405*...).

Seiichi Manyama, Table of n, a(n) for n = 0..4595 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 23.

Cf. A156616, A206622, A206624, A001159 (sigma_4).

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)

{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^3)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 4)-sigma(m, 4))/8*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+15*x+46*x^2+60*x^3+46*x^4+15*x^5+2*x^6)/(1-x^2+x*O(x^n))^4);polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_4(2*n) - sigma_4(n))/8 * x^n/n ), where sigma_4(n) is the sum of 4th powers of divisors of n (A001159).
Inverse Euler transform has g.f.: x*(2 + 15*x + 46*x^2 + 60*x^3 + 46*x^4 + 15*x^5 + 2*x^6)/(1-x^2)^4.
a(n) ~ (93*Zeta(5))^(59/600) * exp(5/4 * (93*Zeta(5)/2)^(1/5) * n^(4/5) + Zeta'(-3)) / (2^(59/100) * sqrt(5*Pi) * n^(359/600)), where Zeta(5) = A013663, Zeta'(-3) = A259068. - Vaclav Kotesovec, Aug 19 2015

A284900 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^4.

Original entry on oeis.org

1, 15, 82, 239, 626, 1230, 2402, 3823, 6643, 9390, 14642, 19598, 28562, 36030, 51332, 61167, 83522, 99645, 130322, 149614, 196964, 219630, 279842, 313486, 391251, 428430, 538084, 574078, 707282, 769980, 923522, 978671, 1200644, 1252830, 1503652, 1587677

Offset: 1

Seiichi Manyama, Apr 05 2017

Multiplicative because this sequence is the Dirichlet convolution of A000583 and A062157 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), this sequence (k=4), A284926 (k=5), A284927 (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).

Cf. A000583, A013663, A062157, A386729.

Table[Sum[(-1)^(n/d + 1)*d^4, {d, Divisors[n]}], {n, 50}] (* Indranil Ghosh, Apr 05 2017 *)
f[p_, e_] := (p^(4*e + 4) - 1)/(p^4 - 1); f[2, e_] := (7*2^(4*e + 1) + 1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^4); \\ Indranil Ghosh, Apr 05 2017

from sympy import divisors
print([sum([(-1)**(n//d + 1)*d**4 for d in divisors(n)]) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017

G.f.: Sum_{k>=1} k^4*x^k/(1 + x^k). - Ilya Gutkovskiy, Apr 07 2017
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (7*2^(4*e+1)+1)/15, and a(p^e) = (p^(4*e+4) - 1)/(p^4 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = 3*zeta(5)/16 = 0.194423... . (End)

A351267 Sum of the 4th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 17, 82, 17, 626, 1394, 2402, 17, 82, 10642, 14642, 1394, 28562, 40834, 51332, 17, 83522, 1394, 130322, 10642, 196964, 248914, 279842, 1394, 626, 485554, 82, 40834, 707282, 872644, 923522, 17, 1200644, 1419874, 1503652, 1394, 1874162, 2215474, 2342084, 10642, 2825762, 3348388

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^4 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 17; a(4) = Sum_{d|4} d^4 * mu(d)^2 = 1^4*1 + 2^4*1 + 4^4*0 = 17.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013663.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), this sequence (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^4); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = sumdiv(n, d, if (issquarefree(d), d^4)); \\ Michel Marcus, Feb 06 2022

a(n) = Sum_{d|n} d^4 * mu(d)^2.
G.f.: Sum_{k>=1} mu(k)^2 * k^4 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^4. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(2)) = 0.126075... . - Amiram Eldar, Nov 10 2022

A351601 a(n) = n^3 * Sum_{d^2|n} 1 / d^3.

Original entry on oeis.org

1, 8, 27, 72, 125, 216, 343, 576, 756, 1000, 1331, 1944, 2197, 2744, 3375, 4672, 4913, 6048, 6859, 9000, 9261, 10648, 12167, 15552, 15750, 17576, 20412, 24696, 24389, 27000, 29791, 37376, 35937, 39304, 42875, 54432, 50653, 54872, 59319, 72000, 68921, 74088, 79507, 95832

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

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

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), this sequence (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013663.

f[p_, e_] := p^3*(p^(3*e) - p^(3*Floor[(e - 1)/2]))/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^3*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^3))); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^3*(p^(3*e) - p^(3*floor((e-1)/2)))/(p^3 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(5)/4 = 0.259231... . - Amiram Eldar, Nov 13 2022

A352032 Sum of the 4th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 82, 1, 1, 82, 626, 1, 82, 1, 2402, 707, 1, 1, 6643, 1, 626, 2483, 14642, 1, 82, 626, 28562, 6643, 2402, 1, 51332, 1, 1, 14723, 83522, 3027, 6643, 1, 130322, 28643, 626, 1, 196964, 1, 14642, 57893, 279842, 1, 82, 2402, 391251, 83603, 28562, 1, 538084, 15267

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 626; a(10) = Sum_{d|10, d<10, d odd} d^4 = 1^4 + 5^4 = 626.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), this sequence (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013663, A051001.

f[2, e_] := 1; f[p_, e_] := (p^(4*e+4) - 1)/(p^4 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^4, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^4 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A051001(n) - n^4*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)-1)/10 = 0.0036927755... . (End)

A352050 Sum of the 4th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 16, 81, 256, 625, 1312, 2401, 4096, 6642, 10016, 14641, 20992, 28561, 38432, 51331, 65536, 83521, 106288, 130321, 160256, 196963, 234272, 279841, 335872, 391250, 456992, 538083, 614912, 707281, 821312, 923521, 1048576, 1200643, 1336352, 1503651, 1700608, 1874161

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^4 * Sum_{d|10, d<10, d odd} 1 / d^4 = 10^4 * (1/1^4 + 1/5^4) = 10016.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), this sequence (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013663, A051001.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^4, d = select(`<`,numtheory:-divisors(m),n))
end proc:map(f, [$1..40]); # Robert Israel, Apr 03 2023

A352050[n_]:=DivisorSum[n,1/#^4&,#A352050,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-4, n/2^IntegerExponent[n, 2]] * n^4 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^4 * sigma(n >> valuation(n, 2), -4) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^4 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^4 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051001(n) * A006519(n)^4 - A000035(n).

Sum_{k=1..n} a(k) = c * n^5 / 5, where c = 31*zeta(5)/32 = 1.00452376... . (End)

A073570 G.f.: Sum_{n >= 1} x^n/(1-x^n)^5.

Original entry on oeis.org

1, 6, 16, 41, 71, 147, 211, 371, 511, 791, 1002, 1547, 1821, 2596, 3146, 4247, 4846, 6627, 7316, 9681, 10852, 13657, 14951, 19427, 20546, 25577, 27916, 34096, 35961, 44912, 46377, 56607, 59922, 70896, 74096, 90278, 91391, 108591, 113766, 133421

Offset: 1

Vladeta Jovovic, Aug 31 2002

nonn

Inverse Moebius transform of pentatope numbers. - Jonathan Vos Post, Mar 31 2006

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

Cf. A000005, A000332, A007437, A013663, A059358, A101289.

Cf. A000203, A001157, A001158, A001159.

Table[(DivisorSigma[4,n]+6*DivisorSigma[3,n]+11*DivisorSigma[2,n]+ 6*DivisorSigma[ 1,n])/24,{n,40}] (* Harvey P. Dale, Aug 08 2013 *)

a(n) = sumdiv(n, d, binomial(d+3, 4)); \\ Seiichi Manyama, Apr 19 2021

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021

a(n) = my(f = factor(n)); (sigma(f, 4) + 6*sigma(f, 3) + 11*sigma(f, 2) + 6*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

a(n) = (1/24) * (sigma_4(n) + 6*sigma_3(n) + 11*sigma_2(n) + 6*sigma_1(n)).
a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)*(d+4)/24 = Sum_{d|n} C(d+3,4) = Sum_{d|n} A000332(d+3). - Jonathan Vos Post, Mar 31 2006. Corrected by Joshua Zucker, May 04 2007
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-4) + 6*zeta(s-3) + 11*zeta(s-2) + 6*zeta(s-2)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007

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cp's OEIS Frontend

A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

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A244667 Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^2) where H(n) is the n-th harmonic number.

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A008382 a(n) = floor(n/5)*floor((n+1)/5)*floor((n+2)/5)*floor((n+3)/5)*floor((n+4)/5).

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A206623 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^3).

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A284900 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^4.

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A351267 Sum of the 4th powers of the squarefree divisors of n.

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A351601 a(n) = n^3 * Sum_{d^2|n} 1 / d^3.

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A008382 a(n) = floor(n/5)floor((n+1)/5)floor((n+2)/5)floor((n+3)/5)floor((n+4)/5).