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Numbers k that are composite and not a powers of a prime k such that sopf^{h+1}(k) divides sopf^{h}(k), with sopf^{0}(k)=k, for h=0..A321944(k)-1, where sopf^{h} is the h-th iteration of sopf and sopf = A008472.
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%I #28 Jun 22 2024 15:47:59

%S 528,1056,1275,1584,2112,2275,2565,3168,3213,3825,3850,3861,4224,4590,

%T 4752,5152,5808,6336,6375,6688,7072,7695,7700,8448,9065,9180,9504,

%U 9639,10304,10878,11328,11375,11475,11583,11616,12672,12825,13376,13770,14144,14256,15400,15925,16709,16896

%N Numbers k that are composite and not a powers of a prime k such that sopf^{h+1}(k) divides sopf^{h}(k), with sopf^{0}(k)=k, for h=0..A321944(k)-1, where sopf^{h} is the h-th iteration of sopf and sopf = A008472.

%e For k = 11475 = 3^3 * 5^2 * 17, sopf(k)=25 divides k and sopf(sopf(k))=5 divides sopf(k).

%p f := proc (n)

%p add(d, d = numtheory[factorset](n))

%p end proc:

%p h := proc (n)

%p option remember;

%p if isprime(n) then

%p 1

%p else

%p 1+h(convert(numtheory[factorset](n), `+`)) end if:

%p end proc:

%p checkDivisibility := proc (n)

%p local k, fk, fk1, result:

%p result := true:

%p fk := n;

%p for k from 0 to h(n)-1 do

%p fk1 := f(fk);

%p if fk1 = 0 or `mod`(fk, fk1) <> 0 then

%p result := false:

%p break:

%p end if:

%p fk := fk1:

%p end do:

%p return result:

%p end proc:

%p g := proc (n)

%p nops(numtheory[factorset](n)):

%p end proc:

%p findNumbers := proc (upper_limit)

%p local n, results:

%p results := []:

%p for n from 2 to upper_limit do

%p if checkDivisibility(n) and 2 <= g(n) then

%p results := [op(results), n]:

%p end if:

%p end do:

%p return results:

%p end proc:

%p upper_limit := 10000:

%p numbers := findNumbers(upper_limit);

%t s[n_] := DivisorSum[n, # &, PrimeQ[#] &]; q[n_] := !PrimePowerQ[n] && AllTrue[Ratios@ Reverse@ FixedPointList[s, n], IntegerQ]; Select[Range[2, 17000], q] (* _Amiram Eldar_, May 30 2024 *)

%Y Cf. A008472 (sopf), A321944.

%K nonn

%O 1,1

%A _Rafik Khalfi_, May 30 2024