A152454 Irregular triangle in which row n lists the numbers whose proper divisors sum to n.

4, 9, 6, 25, 8, 10, 49, 15, 14, 21, 121, 27, 35, 22, 169, 16, 33, 12, 26, 39, 55, 289, 65, 77, 34, 361, 18, 51, 91, 20, 38, 57, 85, 529, 95, 119, 143, 46, 69, 133, 28, 115, 187, 841, 32, 125, 161, 209, 221, 58, 961, 45, 87, 247, 62, 93, 145, 253, 24, 155, 203, 299, 323, 1369

Offset: 2

Author

T. D. Noe, Dec 05 2008

Keywords

• nonn
• tabf

Comments

In an aliquot sequence, all numbers in row n can be predecessors of n. This sequence is a permutation of the composite numbers; number k appears in row A001065(k). We start with n=2 because every prime would be in row 1. Note that row 2 is empty -- as are all the rows listed in A005114. Row n contains A048138(n) numbers. When n is prime, the largest number in row n+1 is n^2. When n>7 is odd, the largest number in row n is less than ((n-1)/2)^2 and (if a strong form of the Goldbach conjecture is true) has the form pq, with primes p

In row n, the first term is A070015(n), and the last term is A135244(n). - Michel Marcus, Nov 11 2014

The first row with several terms is row(6), where the difference between extreme terms is 25-6=19. The next row with a smaller difference is row(13) with a difference 35-27=8. And the next one is row(454) with a difference 602-596=6. Is there a next row with a smaller difference? - Michel Marcus, Nov 11 2014

Examples

			Irregular triangle starts:
  ; (empty row at n=2)
  4;
  9;
  ; (empty row at n=5)
  6, 25;
  8;
  10, 49;
  15;
  14;
  21;
  121;
  27, 35;
  22, 169;
  16, 33;
  12, 26;
  39, 55;
  289;
  ...
		

Links

• T. D. Noe, Rows n=2..1000 of triangle, flattened
• Michel Marcus, Rows n=2..1000 of triangle, not flattened

Crossrefs

Cf. A001065, A005114, A048138.

Cf. A070015, A135244, A135245.

Programs

Maple

N:= 100: # for rows 2 to N, flattened
for s from 2 to N do B[s]:= NULL od:
for k from 1 to N^2 do
  if not isprime(k) then
    s:= numtheory:-sigma(k)-k;
    if s <= N then
       B[s]:= B[s],k;
    fi
  fi
od:
seq(B[s],s=2..N); # Robert Israel, Nov 11 2014

Mathematica

nn=100; s=Table[{},{nn}]; Do[k=DivisorSigma[1,n]-n; If[1
				

PARI

row(n) = select(x->(sigma(x)-x)==n, [1..n^2]); \\ Michel Marcus, Feb 25 2025

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