

A152454


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


13



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
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OFFSET

2,1


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 ((n1)/2)^2 and (if a strong form of the Goldbach conjecture is true) has the form pq, with primes p<q and p+q=n1.
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 256=19. The next row with a smaller difference is row(13) with a difference 3527=8. And the next one is row(454) with a difference 602596=6. Is there a next row with a smaller difference?  Michel Marcus, Nov 11 2014


LINKS

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


EXAMPLE

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;
...


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<k<=nn, AppendTo[s[[k]], n]], {n, nn^2}]; Flatten[s]


CROSSREFS

Cf. A001065, A005114, A048138.
Cf. A070015, A135244, A135245.
Sequence in context: A095065 A061207 A140694 * A287932 A074767 A016097
Adjacent sequences: A152451 A152452 A152453 * A152455 A152456 A152457


KEYWORD

nonn,tabf


AUTHOR

T. D. Noe, Dec 05 2008


STATUS

approved



