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

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

0,7

Comments

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

Links

Table of n, a(n) for n=0..48.

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).

Formula

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

Mathematica

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 *)

Prog

(Maxima) 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 */

Crossrefs

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.

Sequence in context: A289275 A316758 A316750 * A208742 A323395 A326079

Adjacent sequences: A008379 A008380 A008381 * A008383 A008384 A008385

Keyword

nonn

Author

N. J. A. Sloane

Status

approved