A008881 - OEIS

%I #37 Jan 10 2023 01:48:48

%S 0,0,0,0,0,0,1,2,4,8,16,32,64,96,144,216,324,486,729,972,1296,1728,

%T 2304,3072,4096,5120,6400,8000,10000,12500,15625,18750,22500,27000,

%U 32400,38880,46656,54432,63504,74088,86436,100842,117649,134456,153664,175616,200704

%N a(n) = Product_{j=0..5} floor((n+j)/6).

%C For n >= 6, a(n) is the maximal product of 6 positive integers with sum n. - _Wesley Ivan Hurt_, Jun 29 2022

%C The maximal product of k positive variables when their sum is equal to s is obtained when each term = s/k; hence, a(6m) = m^6 (A001014). - _Bernard Schott_, Jul 28 2022

%H G. C. Greubel, <a href="/A008881/b008881.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,5,-10,5,0,0,0,-10,20,-10,0,0,0,10,-20,10,0,0,0,-5,10,-5,0,0,0,1,-2,1).

%F Sum_{n>=6} 1/a(n) = 1 + zeta(6). - _Amiram Eldar_, Jan 10 2023

%p seq( mul( floor((n+i)/6), i=0..5 ), n=0..80);

%t Product[Floor[(Range[51]+j-2)/6], {j,6}] (* _G. C. Greubel_, Sep 13 2019 *)

%o (PARI) vector(50, n, prod(j=0,5, (n+j)\6) ) \\ _G. C. Greubel_, Sep 13 2019

%o (Magma) [(&*[Floor((n+j)/6): j in [0..5]]): n in [0..50]]; // _G. C. Greubel_, Sep 13 2019

%o (Sage) [product(floor((n+j)/6) for j in (0..5)) for n in (0..50)] # _G. C. Greubel_, Sep 13 2019

%o (GAP) List([0..50], n-> Product([0..5], j-> Int((n+j)/6))); # _G. C. Greubel_, Sep 13 2019

%Y Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), this sequence (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

%Y Cf. A001014 (6th power), A008588 (multiples of 6), A013664.

%K nonn,easy

%O 0,8

%A _N. J. A. Sloane_