A211221 -id:A211221 - cp's OEIS frontend

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

Paolo P. Lava, Apr 11 2012

nonn

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

			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.

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

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

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

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 *)

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

A353306 Numbers k such that 1 is in the transitive closure of the map x -> A353313(x) when starting iterating from x=k.

Original entry on oeis.org

1, 3, 4, 9, 12, 27, 36, 81, 108, 193, 243, 324, 346, 436, 579, 729, 972, 1038, 1308, 1522, 1737, 1867, 2187, 2353, 2539, 2916, 3114, 3493, 3924, 4234, 4566, 5211, 5601, 5824, 6286, 6561, 7059, 7617, 8748, 9342, 9446, 9709, 10479, 10886, 11756, 11772, 12702, 13698, 13772, 14792, 14806, 15633, 15745, 16184, 16803, 17003

Offset: 1

Antti Karttunen, Apr 13 2022

nonn

Positions of ones in A353305.

Subsequences: A000244, A003946, A211221.

A353313(n) = { my(r=(n%3)); if(!r,n/3,((5*((n-r)/3)) + r + 3)); };
A353305(n) = { my(visited = Map(), m=0); for(j=1, oo, n = A353313(n); if(!m, m=n, m=min(m,n)); if(mapisdefined(visited, n), return(m), mapput(visited, n, j))); };
isA353306(n) = (1==A353305(n));

cp's OEIS Frontend

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

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