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.
f := proc(n) option remember; if n = 0 then 150; else sigma(f(n-1))-f(n-1); fi; end:
FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 150] // Most (* Jean-François Alcover, Mar 28 2020 *)
a(n,a=150)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018
a(12) = 7: 12 is divisible by 1,2,3,4 and 6 so sigma(12)=16; 16 is divisible by 1,2,4 and 8 so sigma(16)=15; 15 is divisible by 1,3 and 5 so sigma(15)=9; 9 is divisible by 1 and 3 so sigma(9)=4; 4 is divisible by 1 and 2 so sigma(4)=3; 3 is divisible only by 1 so sigma(3)=1; 1 is not divisible by anything less than 1 so sigma(1)=0. The aliquot sequence is therefore 16, 15, 9, 4, 3, 1, 0 which is 7 elements long. Therefore a(12) = 7.
f[n_]:=Plus@@Divisors[n]-n;lst2={};Do[lst={};a=k;Do[b=a;a=f[a];AppendTo[lst,a];If[a==0||a==b,Break[]],{n,7!}];AppendTo[lst2,Length[lst]],{k,5!}];lst2 (* Vladimir Joseph Stephan Orlovsky, Apr 24 2010 *)
(* This script is not suitable for a large number of terms *) 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[n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] &, All] == 1; selQ /@ Range[1000]; A131884 (* Jean-François Alcover, Sep 10 2015 *)
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
FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 46758, 100] (* Jean-François Alcover, Mar 28 2020 *)
a(n,a=46758)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018
f := proc(n) option remember; if n = 0 then 180; else sigma(f(n-1))-f(n-1); fi; end:
FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 180] // Most (* Jean-François Alcover, Mar 28 2020 *)
a(n,a=180)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018
FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 564, 100] (* Jean-François Alcover, Mar 28 2020 *)
a(n, a=564)={for(i=1, n, a=sigma(a)-a); a} \\ M. F. Hasler, Feb 24 2018
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] &]
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