%I #24 Nov 15 2013 04:17:56

%S 25,95,119,143,417,445,565,608,650,652,675,685,783,790,909,913

%N Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number.

%C There are many numbers whose aliquot sequences have not yet been completely computed, so this sequence is not fully known. In particular, 276 may, perhaps, be an element of this sequence, although this is very unlikely.

%C Numbers less than 1000 whose aliquot sequence is not known that could possibly be in this sequence are: 276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996. - _Robert Price_, Jun 03 2013

%D No number terminates at 28, the second perfect number.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AspiringNumber.html">Aspiring Number</a>

%e The divisors of 95 less than itself are 1, 5 and 19. They sum to 25. The divisors of 25 less than itself are 1 and 5. They sum to 6, which is perfect.

%t perfectQ[n_] := DivisorSigma[1, n] == 2*n; maxAliquot = 10^45; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = False; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] &, All] // perfectQ; Reap[For[k = 1, k < 1000, k++, If[! perfectQ[k] && selQ[k], Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Nov 15 2013 *)

%Y Cf. A080907, A115350.

%K hard,nice,nonn

%O 1,1

%A _Tanya Khovanova_ and Alexey Radul, Aug 14 2001

%E a(13)-a(16) from _Robert Price_, Jun 03 2013