A119348 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct divisors of n.

1, 1, 2, 3, 1, 3, 4, 1, 2, 3, 4, 5, 6, 7, 1, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 3, 4, 9, 10, 12, 13, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 1, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Offset

1,3

Comments

Row n contains A119347(n) terms. In row n the first term is 1 and the last term is sigma(n) (=sum of the divisors of n =A000203(n)). If row n contains all numbers from 1 to sigma(n), then n is called a practical number (A005153).

Links

T. D. Noe, Rows n=1..100, flattened

Example

Row 5 is 1,5,6, the possible sums obtained from the divisors 1 and 5 of 5.
Triangle starts:
1;
1,2,3;
1,3,4;
1,2,3,4,5,6,7;
1,5,6;
1,2,3,4,5,6,7,8,9,10,11,12;
1,7,8;
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
1,3,4,9,10,12,13;

Maple

with(numtheory): with(linalg): sums:=proc(n) local dl, t: dl:=convert(divisors(n), list): t:=tau(n): {seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)} end: for n from 1 to 12 do sums(n) od; # yields sequence in triangular form

Mathematica

row[n_] := Union[Total /@ Subsets[Divisors[n]]] // Rest;
Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Aug 06 2024 *)

Crossrefs

Cf. A000203, A005153, A119347.

Sequence in context: A238793 A240677 A030306 * A282935 A354223 A181974

Adjacent sequences: A119345 A119346 A119347 * A119349 A119350 A119351

Keyword

nonn,tabf

Author

Emeric Deutsch, May 15 2006

Status

approved