A193280 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct proper divisors of n.

0, 1, 1, 1, 2, 3, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 3, 4, 1, 2, 3, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 1, 2, 3, 7, 8, 9, 10, 1, 3, 4, 5, 6, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 1

Author

Michael Engling, Jul 20 2011

Keywords

• nonn
• tabf

Comments

Row n > 1 contains A193279(n) terms. In row n the first term is 1 and the last term is sigma(n) - n (= A000203(n) - n). Row 1 contains 0 because 1 has no proper divisors.

Examples

			Row 10 is 1,2,3,5,6,7,8 the possible sums obtained from the proper divisors 1, 2, and 5 of 10.
Triangle starts:
  0;
  1;
  1;
  1,2,3;
  1;
  1,2,3,4,5,6;
  1;
  1,2,3,4,5,6,7;
  1,3,4;
  1,2,3,5,6,7,8;
  1;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16;
		

Links

• Nathaniel Johnston, Rows 1..150, flattened

Crossrefs

Cf. A119347, A119348, A193279.

Programs

Maple

with(linalg): print(0); for n from 2 to 12 do dl:=convert(numtheory[divisors](n) minus {n}, list): t:=nops(dl): print(op({seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)})): od: # Nathaniel Johnston, Jul 23 2011