A063769 Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number.

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

Offset: 1

Author

Tanya Khovanova and Alexey Radul, Aug 14 2001

Keywords

• hard
• nice
• nonn

Comments

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.

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

Examples

			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.
		

References

• No number terminates at 28, the second perfect number.

Links

• Eric Weisstein's World of Mathematics, Aspiring Number

Crossrefs

Cf. A080907, A115350.

Programs

Mathematica

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 *)

Extensions

a(13)-a(16) from Robert Price, Jun 03 2013