A330437 Length of trajectory of n under the map n -> n - 1 + n/gpf(n) or 0 if no fixed point is reached, where gpf(n) is the greatest prime factor of n.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 5, 4, 2, 1, 4, 1, 2, 4, 4, 3, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 1, 3, 2, 4, 3, 2, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 2, 2, 1, 4, 2, 2, 1, 4, 2, 3, 2, 4, 1, 2, 2, 4, 4, 4, 3, 2, 1, 3, 2, 4

Offset: 1

Author

Elijah Beregovsky, Feb 16 2020

Keywords

• nonn

Comments

The table of trajectories of n under is given in A329288.

All fixed points, besides 1, are prime.

Conjecture: every number appears in the sequence infinitely many times.

Conjecture: all terms are nonzero, i.e., every trajectory eventually reaches a prime.

Examples

			For n = 26 the trajectory is (26, 27, 35, 39, 41) so a(26) = 5.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006530 (greatest prime factor), A329288, A330704 (greedy inverse).

Cf. A270807, A120685, A331410.

Programs

Maple

g:= n -> n - 1 + n/max(numtheory:-factorset(n)):
f:= proc(n) option remember;
    if isprime(n) then 1 else 1+ procname(g(n)) fi
end proc:
f(1):= 1:
map(f, [$1..200]); # Robert Israel, May 01 2020

Mathematica

Clear[f,it,order,seq]; f[n_]:=f[n]=n-1+n/FactorInteger[n][[-1]][[1]]; it[k_,n_]:=it[k,n]=f[it[k,n-1]]; it[k_,1]=k; order[n_]:=order[n]=SelectFirst[Range[1,100], it[n,#]==it[n,#+1]&]; Print[order/@Range[1,100]];

PARI

apply( {a(n,c=1)=n>1&&while(nM. F. Hasler, Feb 19 2020

Formula

a(p) = 1 for any prime number p.