

A211220


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


2



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
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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



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);
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:
# 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(ni*j, i1), j=0..n/i))))
end:
a:= n> b(n$2):


MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, 0, If[i<1, Infinity, Max[Table[ DivisorSigma[1, i]*j + b[ni*j, i1], {j, 0, n/i}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* JeanFrançois Alcover, Feb 16 2017, after Alois P. Heinz *)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



