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

1, 3, 4, 7, 8, 12, 13, 15, 16, 19, 20, 28, 29, 31, 32, 35, 36, 40, 41, 43, 44, 47, 48, 60, 61, 63, 64, 67, 68, 72, 73, 75, 76, 79, 80, 91, 92, 94, 95, 98, 99, 103, 104, 106, 107, 110, 111, 124, 125, 127, 128, 131, 132, 136, 137, 139, 140, 143, 144, 168, 169

Offset

1,2

Comments

For n equal to 1, 2, 3, 4, 6, 8, 12, 24, 30, 36, etc. the maximum value is equal to sigma(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Example

For n=10 the partition (4,6) gives sigma(4)+sigma(6)= 7 + 12 = 19 that is the maximum value that can be reached.
For n=21 the partitions (1,8,12), (3,6,12) and (1,2,6,12) give:
sigma(1)+sigma(8)+sigma(12)= 1 + 15 + 28 = 44;
sigma(3)+sigma(6)+sigma(12)= 4 + 12 + 28 = 44;
sigma(1)+sigma(2)+ sigma(6)+sigma(12)= 1 + 3 + 12 + 28 = 44
that is the maximum value that can be reached.

Maple

with(numtheory); with(combinat);
A211220:=proc(q)
local b, c, i, j, k, m, n, t;
for n from 1 to q do
  k:=partition(n); b:=numbpart(n); m:=0;
  for i from 1 to b do
    c:=nops(k[i]); t:=0;
    for j from 1 to c do t:=t+sigma(k[i][j]); od; if t>m then m:=t; fi; od;
  print(m);
od; end:
A211220(100);
# second Maple program:
with(numtheory):
b:= proc(n, i) option remember; `if`(n=0, 0, `if`(i<1,
      -infinity, max(seq(sigma(i)*j+b(n-i*j, i-1), j=0..n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..70);  # Alois P. Heinz, May 30 2013

Mathematica

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 *)

Crossrefs

Cf. A000203, A085884, A211217-A211219, A211221.

Sequence in context: A014602 A078823 A045615 * A051201 A026449 A286904

Adjacent sequences: A211217 A211218 A211219 * A211221 A211222 A211223

Keyword

nonn

Author

Paolo P. Lava, Apr 11 2012

Extensions

Extended beyond a(47) by Alois P. Heinz, May 30 2013

Status

approved