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A152454 Irregular triangle in which row n lists the numbers whose proper divisors sum to n. 13


%S 4,9,6,25,8,10,49,15,14,21,121,27,35,22,169,16,33,12,26,39,55,289,65,

%T 77,34,361,18,51,91,20,38,57,85,529,95,119,143,46,69,133,28,115,187,

%U 841,32,125,161,209,221,58,961,45,87,247,62,93,145,253,24,155,203,299,323,1369

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

%C 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 In row n, the first term is A070015(n), and the last term is A135244(n). - _Michel Marcus_, Nov 11 2014

%C 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 T. D. Noe, <a href="/A152454/b152454.txt">Rows n=2..1000 of triangle, flattened</a>

%H Michel Marcus, <a href="/A152454/a152454.txt">Rows n=2..1000 of triangle, not flattened</a>

%e Irregular triangle starts:

%e ; (empty row at n=2)

%e 4;

%e 9;

%e ; (empty row at n=5)

%e 6, 25;

%e 8;

%e 10, 49;

%e 15;

%e 14;

%e 21;

%e 121;

%e 27, 35;

%e 22, 169;

%e 16, 33;

%e 12, 26;

%e 39, 55;

%e 289;

%e ...

%p N:= 100: # for rows 2 to N, flattened

%p for s from 2 to N do B[s]:= NULL od:

%p for k from 1 to N^2 do

%p if not isprime(k) then

%p s:= numtheory:-sigma(k)-k;

%p if s <= N then

%p B[s]:= B[s],k;

%p fi

%p fi

%p od:

%p seq(B[s],s=2..N); # _Robert Israel_, Nov 11 2014

%t nn=100; s=Table[{},{nn}]; Do[k=DivisorSigma[1,n]-n; If[1<k<=nn, AppendTo[s[[k]],n]], {n,nn^2}]; Flatten[s]

%Y Cf. A001065, A005114, A048138.

%Y Cf. A070015, A135244, A135245.

%K nonn,tabf

%O 2,1

%A _T. D. Noe_, Dec 05 2008

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Last modified August 5 16:12 EDT 2021. Contains 346479 sequences. (Running on oeis4.)