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

%I A152454 #23 Feb 25 2025 11:56:44
%S A152454 4,9,6,25,8,10,49,15,14,21,121,27,35,22,169,16,33,12,26,39,55,289,65,
%T A152454 77,34,361,18,51,91,20,38,57,85,529,95,119,143,46,69,133,28,115,187,
%U A152454 841,32,125,161,209,221,58,961,45,87,247,62,93,145,253,24,155,203,299,323,1369
%N A152454 Irregular triangle in which row n lists the numbers whose proper divisors sum to n.
%C A152454 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<q and p+q=n-1.
%C A152454 In row n, the first term is A070015(n), and the last term is A135244(n). - _Michel Marcus_, Nov 11 2014
%C A152454 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
%H A152454 T. D. Noe, <a href="/A152454/b152454.txt">Rows n=2..1000 of triangle, flattened</a>
%H A152454 Michel Marcus, <a href="/A152454/a152454.txt">Rows n=2..1000 of triangle, not flattened</a>
%e A152454 Irregular triangle starts:
%e A152454   ; (empty row at n=2)
%e A152454   4;
%e A152454   9;
%e A152454   ; (empty row at n=5)
%e A152454   6, 25;
%e A152454   8;
%e A152454   10, 49;
%e A152454   15;
%e A152454   14;
%e A152454   21;
%e A152454   121;
%e A152454   27, 35;
%e A152454   22, 169;
%e A152454   16, 33;
%e A152454   12, 26;
%e A152454   39, 55;
%e A152454   289;
%e A152454   ...
%p A152454 N:= 100: # for rows 2 to N, flattened
%p A152454 for s from 2 to N do B[s]:= NULL od:
%p A152454 for k from 1 to N^2 do
%p A152454   if not isprime(k) then
%p A152454     s:= numtheory:-sigma(k)-k;
%p A152454     if s <= N then
%p A152454        B[s]:= B[s],k;
%p A152454     fi
%p A152454   fi
%p A152454 od:
%p A152454 seq(B[s],s=2..N); # _Robert Israel_, Nov 11 2014
%t A152454 nn=100; s=Table[{},{nn}]; Do[k=DivisorSigma[1,n]-n; If[1<k<=nn, AppendTo[s[[k]],n]], {n,nn^2}]; Flatten[s]
%o A152454 (PARI) row(n) = select(x->(sigma(x)-x)==n, [1..n^2]); \\ _Michel Marcus_, Feb 25 2025
%Y A152454 Cf. A001065, A005114, A048138.
%Y A152454 Cf. A070015, A135244, A135245.
%K A152454 nonn,tabf
%O A152454 2,1
%A A152454 _T. D. Noe_, Dec 05 2008