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

Author

Emeric Deutsch, May 15 2006

Keywords

• nonn
• tabf

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).

Examples

			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;
		

Links

• T. D. Noe, Rows n=1..100, flattened

Crossrefs

Cf. A000203, A005153, A119347.

Programs

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 *)