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A211220 For any partition of n consider the sum of the sigma of each element. Sequence gives the maximum of such values.

%I A211220 #18 Jun 11 2025 14:57:38
%S A211220 1,3,4,7,8,12,13,15,16,19,20,28,29,31,32,35,36,40,41,43,44,47,48,60,
%T A211220 61,63,64,67,68,72,73,75,76,79,80,91,92,94,95,98,99,103,104,106,107,
%U A211220 110,111,124,125,127,128,131,132,136,137,139,140,143,144,168,169
%N A211220 For any partition of n consider the sum of the sigma of each element. Sequence gives the maximum of such values.
%C A211220 For n equal to 1, 2, 3, 4, 6, 8, 12, 24, 30, 36, etc. the maximum value is equal to sigma(n).
%H A211220 Alois P. Heinz, <a href="/A211220/b211220.txt">Table of n, a(n) for n = 1..1000</a>
%e A211220 For n=10 the partition (4,6) gives sigma(4)+sigma(6)= 7 + 12 = 19 that is the maximum value that can be reached.
%e A211220 For n=21 the partitions (1,8,12), (3,6,12) and (1,2,6,12) give:
%e A211220 sigma(1)+sigma(8)+sigma(12)= 1 + 15 + 28 = 44;
%e A211220 sigma(3)+sigma(6)+sigma(12)= 4 + 12 + 28 = 44;
%e A211220 sigma(1)+sigma(2)+ sigma(6)+sigma(12)= 1 + 3 + 12 + 28 = 44
%e A211220 that is the maximum value that can be reached.
%p A211220 with(numtheory):
%p A211220 b:= proc(n, i) option remember; `if`(n=0, 0, `if`(i<1,
%p A211220       -infinity, max(seq(sigma(i)*j+b(n-i*j, i-1), j=0..n/i))))
%p A211220     end:
%p A211220 a:= n-> b(n$2):
%p A211220 seq(a(n), n=1..70);  # _Alois P. Heinz_, May 30 2013
%t A211220 b[n_, i_] := b[n, i] = If[n==0, 0, If[i<1, -Infinity, Max[Table[ DivisorSigma[1, i]*j + b[n-i*j, i-1], {j, 0, n/i}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* _Jean-François Alcover_, Feb 16 2017, after _Alois P. Heinz_ *)
%Y A211220 Cf. A000203, A085884, A211217-A211219, A211221.
%K A211220 nonn
%O A211220 1,2
%A A211220 _Paolo P. Lava_, Apr 11 2012
%E A211220 Extended beyond a(47) by _Alois P. Heinz_, May 30 2013