cp's OEIS Frontend

A211221 For any partition of n consider the product of the sigma of each element. Sequence gives the maximum of such values.

%I A211221 #27 Apr 10 2024 10:44:57
%S A211221 1,3,4,9,12,27,36,81,108,243,324,729,972,2187,2916,6561,8748,19683,
%T A211221 26244,59049,78732,177147,236196,531441,708588,1594323,2125764,
%U A211221 4782969,6377292,14348907,19131876,43046721,57395628,129140163,172186884,387420489,516560652
%N A211221 For any partition of n consider the product of the sigma of each element. Sequence gives the maximum of such values.
%H A211221 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).
%F A211221 For n>1, a(n) = 3^n/2 for n even and a(n) = 4*3^(n-3)/2 for n odd.
%F A211221 For n>3, a(n) = 3*a(n-2). G.f.: x*(1+3*x+x^2)/(1-3*x^2). [_Colin Barker_, Apr 18 2012]
%F A211221 Closed form: a(1)=1, then a(n) = 1/6*(7-(-1)^(n-2))*3^(1/4*(-1)^(n-2))*3^(1/2*(n-2))*27^(1/4) = 3^((2*n+(-1)^n-5)/4)*(7-(-1)^n)/2. [_Paolo P. Lava_, Apr 20 2012]
%e A211221 For n=21 the partition (2,2,2,2,2,2,2,2,2,3) gives sigma(2)^9*sigma(3)=3^9*4=78732 that is the maximum value that can be reached.
%p A211221 with(numtheory); with(combinat);
%p A211221 A211221:=proc(q)
%p A211221 local b,c,i,j,k,m,n,t;
%p A211221 for n from 1 to q do
%p A211221   k:=partition(n); b:=numbpart(n); m:=0;
%p A211221   for i from 1 to b do
%p A211221     c:=nops(k[i]); t:=1;
%p A211221     for j from 1 to c do t:=t*sigma(k[i][j]); od; if t>m then m:=t; fi; od;
%p A211221   print(m);
%p A211221 od; end:
%p A211221 A211221(100)
%t A211221 LinearRecurrence[{0,3},{1,3,4},40] (* _Harvey P. Dale_, Jun 06 2015 *)
%Y A211221 Cf. A000203, A085884, A211217, A211218, A211219, A211220.
%K A211221 nonn,easy
%O A211221 1,2
%A A211221 _Paolo P. Lava_, Apr 13 2012