A390728 Irregular array read by rows: lexicographically earliest sublist of [d: d|k, d < k] that adds up to k and is maximal in length, where k is a semiperfect number.

1, 2, 3, 1, 2, 3, 6, 1, 2, 6, 9, 1, 4, 5, 10, 1, 2, 3, 4, 6, 8, 1, 2, 4, 7, 14, 1, 3, 5, 6, 15, 2, 3, 4, 6, 9, 12, 1, 2, 4, 5, 8, 20, 1, 6, 14, 21, 1, 2, 3, 4, 6, 8, 24, 1, 2, 6, 18, 27, 1, 2, 4, 7, 14, 28, 1, 2, 3, 4, 5, 10, 15, 20, 2, 3, 6, 22, 33, 1, 2, 3, 4, 6, 8, 12, 36

Offset

1,2

Links

Table of n, a(n) for n=1..86.

Example

Array begins:
   6:  1,  2,  3;
  12:  1,  2,  3,  6;
  18:  1,  2,  6,  9;
  20:  1,  4,  5, 10;
  24:  1,  2,  3,  4,  6, 8;
  28:  1,  2,  4,  7, 14;
  30:  1,  3,  5,  6, 15;
  36:  2,  3,  4,  6,  9, 12;
  40:  1,  2,  4,  5,  8, 20;
  42:  1,  6, 14, 21;
  48:  1,  2,  3,  4,  6,  8, 24;
  54:  1,  2,  6, 18, 27;
  56:  1,  2,  4,  7, 14, 28;
  60:  1,  2,  3,  4,  5, 10, 15, 20;
  66:  2,  3,  6, 22, 33;
  72:  1,  2,  3,  4,  6,  8, 12, 36;
  78: 13, 26, 39;
.
Examples where maximality of length is not unique:
k = 48 has also the solution [1, 2, 3, 6, 8, 12, 16].
k = 72 has also the solutions [1, 2, 3, 4, 8, 12, 18, 24], [1, 2, 4, 6, 8, 9, 18, 24].

Maple

with(combinat):
A390728row := proc(n) local S, ls, s, ret;
    if not isA005835(n) then return NULL fi;
    S := subsets(numtheory[divisors](n) minus {n});
    ret := []; ls := 0;
    while not S[finished] do
        s := S[nextvalue]();
        if convert(s, `+`) = n then
            if ls < nops(s) then ret := s; ls := nops(s) fi;
        end if ;
    end do;
ret end proc: seq(A390728row(n), n = 2..78);

Crossrefs

Cf. A005835, A390834 (length of rows).

Sequence in context: A238837 A309940 A081514 * A109206 A292746 A176506

Adjacent sequences: A390725 A390726 A390727 * A390729 A390730 A390731

Keyword

nonn,tabf

Author

Peter Luschny, Nov 25 2025

Status

approved