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

1,4

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.

Links

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

Formula

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

Example

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

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]];

Prog

(PARI) apply( {a(n, c=1)=n>1&&while(n<n+=n/vecmax(factor(n)[, 1])-1, c++); c}, [1..99]) \\ M. F. Hasler, Feb 19 2020

Crossrefs

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

Cf. A270807, A120685, A331410.

Sequence in context: A113309 A348955 A062362 * A338648 A269252 A291150

Adjacent sequences: A330434 A330435 A330436 * A330438 A330439 A330440

Keyword

nonn

Author

Elijah Beregovsky, Feb 16 2020

Status

approved