A334245 Let S = A061228, T = A070229, and F = S o T (where o denotes composition). Then a(n) is the least number in the intersection of the sequences F^k(n) and F^k(S(n)), k>=0, or the intersection of the sequences T(F^k(n)) and T(F^k(S(n))), k>=0, or 0 if these intersections are empty.

12, 12, 12, 15, 12, 30, 15, 21, 15, 77, 21, 77, 30, 77, 30, 77, 30, 57, 77, 30, 77, 91, 77, 77, 77, 77, 35, 105, 77, 133, 77, 91, 77, 60, 77, 111, 57, 77, 77, 133, 77, 165, 91, 65, 91, 147, 60, 105, 65, 70, 65, 165, 77, 105, 105, 91, 105, 180, 133, 231, 133, 91, 105, 105, 105, 234, 105, 147, 91

Offset: 2

Author

Ali Sada and Robert Israel, Apr 19 2020

Keywords

• nonn
• look

Comments

60420 >= a(n) > 0 for n <= 6042, but either a(6043) = 0 or a(6043) > 10^30.

If p is prime, a(p) = a(2*p).

From n = 5, 55, 2, 24, 245, ... begin successive strings of exactly 1, 2, 3, 4, 5, ... identical merging points that are: 15, 105, 12, 77, 713, ... . - Bernard Schott, Jul 04 2020

Examples

			For n=5, 15 = T(F(5)) = T(S(5)) so a(5) = 15.
For n=6, 12 = F(6) = F(S(6)) so a(6) = 12.
For n=10, 15 = T(10) = T(F(10)) so a(10) = 15.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..6042
• A. Sada et al., Will this pattern continue, Discussion in SeqFan mailing list, Apr 2020.
• Bernard and Blandine Schott, Network with terms < 100.

Crossrefs

Cf. A061228, A070229, A335859.

Programs

Maple

S:= x -> x + min(numtheory:-factorset(x)):
T:= x -> x + max(numtheory:-factorset(x)):
f:= proc(n) local s,t,i,j,ct;
  s:= S(n); i:= 0; t:= T(n); j:= 1;
  for ct from 1 to 1000 do
    if s = t and i=j then return s fi;
    if s <= t then
      if i = 0 then s:= T(s); i:= 1
      else s:= S(s); i:= 0
      fi
    elif j = 0 then t:= T(t); j:= 1
    else t:= S(t); j:= 0
    fi
  od;
  0
end proc:
map(f, [$2..100]);

Mathematica

S[x_] := x + FactorInteger[x][[1, 1]];
T[x_] := x + FactorInteger[x][[-1, 1]];
f[n_] := Module[{s = S[n], t = T[n], i = 0, j = 1, ct}, For[ct = 1, ct <= 1000, ct++, If[s == t && i == j, Return[s]]; If[s <= t, If[i == 0, s = T[s]; i = 1, s = S[s]; i = 0], If[j == 0, t = T[t]; j = 1, t = S[t]; j = 0]]]; 0];
f /@ Range[2, 100] (* Jean-François Alcover, Aug 15 2020, after Maple *)

Formula

a(2^m+2) = a(2^m). - Bernard Schott, Jun 27 2020