This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The aspiring number 25, whose aliquot sequence (25 -> 6) terminates in 6, which is a perfect number.
4 is in this set because its aliquot chain is 4->3->1. 6 is not in this set because it is perfect. 25 is not in this set because its aliquot chain is 25->6.
maxAliquot = 10^50; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] & , All] == 1; Select[Range[1, 1100], selQ] (* Jean-François Alcover, Nov 14 2013, updated Sep 10 2015 *)
a(12)=3 since the aliquot sequence starting at 12 is: 12 - 16 - 15 - 9 - 4 - 3. a(95)=6 since the aliquot sequence starting at 95 is: 95 - 25 - 6 - 6 ...
a[n_] := If[n == 1, 1, FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, n] /. {k__, 1, 0, 0} :> {k} // Last];
Array[a, 100] (* Jean-François Alcover, Mar 28 2020 *)
a(24)=55 because the aliquot sequence starting at 24 is: 24 - 36 - 55 - 17 - 1 so the maximum peak of this sequence is 55.
from sympy import divisor_sigma as sigma
def aliquot(n):
alst = []; seen = set(); i = n
while i and i not in seen: alst.append(i); seen.add(i); i = sigma(i) - i
return alst
def aupton(terms): return [max(aliquot(n)) for n in range(1, terms+1)]
print(aupton(66)) # Michael S. Branicky, Jul 11 2021
maxAliquot = 10^45; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] & , All] == 1; Reap[For[k = 1, k < 1100, k++, If[!selQ[k], Print[k]; Sow[k]]]][[2, 1]]
a(5)=15 because the fifth integer whose aliquot sequence terminates by encountering the prime 3 as a member of its trajectory is 15. The complete aliquot sequence generated by iterating the proper divisors of 15 is 15->9->4->3->1->0
s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[275], MemberQ[Trajectory[ # ], 3] &]
a(5)=20 because the fifth integer whose aliquot sequence terminates by encountering the prime 7 as a member of its trajectory is 20. The complete aliquot sequence generated by iterating the proper divisors of 15 is 20->22->14->10->8->7->1->0
s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[275], MemberQ[Trajectory[ # ], 7] &]
a(10)=12 because the tenth integer whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0
s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], Last[Trajectory[ # ]] == 0 &]
a(5)=12 because the fifth composite number whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0
s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], ! PrimeQ[ # ] && Last[Trajectory[ # ]] == 0 &]
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