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.
138 has a transient of length 177 (see Guy's book).
g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; Do[b = Length[ f[n]] - 2; If[b > a, a = b; Print[b]], {n, 275}] (* Robert G. Wilson v, Sep 10 2004 *)
Examples of trajectories: 1, 0. 2, 1, 0. 3, 1, 0. (and similarly for any prime) 4, 3, 1, 0. 5, 1, 0. 6, 6, 6, ... (and similarly for any perfect number) 8, 7, 1, 0. 9, 4, 3, 1, 0. 12, 16, 15, 9, 4, 3, 1, 0. 14, 10, 8, 7, 1, 0. 25, 6, 6, 6, ... 28, 28, 28, ... (the next perfect number) 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0. 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0.
f:=proc(n) local t1, i,j,k; t1:=[n]; for i from 2 to 50 do j:= t1[i-1]; k:=sigma(j)-j; t1:=[op(t1), k]; od: t1; end; # produces trajectory for n # 2nd implementation: A098007 := proc(n) local trac, x; x := n ; trac := [x] ; while true do x := numtheory[sigma](x)-trac[-1] ; if x = 0 then return 1+nops(trac) ; elif x in trac then return nops(trac) ; end if; trac := [op(trac), x] ; end do: end proc: seq(A098007(n), n=1..100) ; # R. J. Mathar, Oct 08 2017
g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; Table[ Length[ f[n]] - 1, {n, 100}] (* Robert G. Wilson v, Sep 10 2004 *)
apply( {A098007(n, t=0)=until(bittest(t,if(n,n=sigma(n)-n)),t+=1<M. F. Hasler, Feb 24 2018, improved Aug 14 2022 thanks to a remark from Jianing Song
from sympy import divisor_sigma as sigma
def a(n, limit=float('inf')):
alst = []; seen = set(); i = n; c = 0
while i and i not in seen and c < limit:
alst.append(i); seen.add(i); i = sigma(i) - i; c += 1
return "NA" if c == limit else len(set(alst + [i]))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 11 2021
(define (A098007 n) (let loop ((visited (list n)) (i 1)) (let ((next (A001065 (car visited)))) (cond ((zero? next) (+ 1 i)) ((member next visited) i) (else (loop (cons next visited) (+ 1 i))))))) (define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry. ;; Antti Karttunen, Nov 01 2017
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
The aliquot sequence of 30 is: 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1. The maximum is 259 which is larger than 30, and larger than the maxima of all the aliquot sequences of the numbers below 30.
g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; seq = {}; a = -1; seq = {}; Do[b = Max[Drop[f[n], 1]]; If[b > a, a = b; AppendTo[seq, n]], {n, 2, 275}]; seq (* after Robert G. Wilson v at A098009 *)
The aliquot sequence of 30 is: 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1. The maximum is 259 which is larger than 30, and larger than the maxima of all the aliquot sequences of the numbers below 30.
g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; rec = {}; a = -1; seq = {}; Do[b = Max[Drop[f[n], 1]];
If[b > a, a = b; AppendTo[rec, b]], {n, 2, 275}] ; rec (* after Robert G. Wilson v at A098009 *)
The unitary aliquot sequence of 134 is: 134, 70, 74, 40, 14, 10, 8, 1. Its length is 8 and it is longer than the unitary aliquot sequences of all the numbers below 134.
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; seq = {}; Do[b = Length[f[n]] - 1; If[b > a, a = b; AppendTo[seq, n]], {n, 10^6}] ; seq (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)
The unitary aliquot sequence of 134 is: 134, 70, 74, 40, 14, 10, 8, 1. Its length is 8 and it is longer than the unitary aliquot sequences of all the numbers below 134.
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = 0; seq = {}; Do[b = Length[f[n]] - 2; If[b > a, a = b; AppendTo[seq, b]], {n, 10^6}]; seq (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)
The unitary aliquot sequence of 174 is: 174, 186, 198, 162, 84, 76, 24, 12, 8, 1. Its maximum is 198 which larger than the maxima of all the aliquot sequences of the numbers below 174.
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; rec = {}; Do[b = Length[f[n]] - 2;
If[b > a, a = b; AppendTo[rec, n]], {n, 10^6}] ; rec (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)
The unitary aliquot sequence of 174 is: 174, 186, 198, 162, 84, 76, 24, 12, 8, 1. Its maximum is 198 which larger than the maxima of all the aliquot sequences of the numbers below 174.
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; rec = {}; Do[b = Length[f[n]] - 2; If[b > a, a = b; AppendTo[rec, b ]], {n, 10^6}] ; rec (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)
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