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

Author

Paolo P. Lava, Apr 11 2012

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Maple

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 *)

Extensions

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