A321944 -id:A321944 - cp's OEIS frontend

A373285 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.

528, 1056, 1275, 1584, 2112, 2275, 2565, 3168, 3213, 3825, 3850, 3861, 4224, 4590, 4752, 5152, 5808, 6336, 6375, 6688, 7072, 7695, 7700, 8448, 9065, 9180, 9504, 9639, 10304, 10878, 11328, 11375, 11475, 11583, 11616, 12672, 12825, 13376, 13770, 14144, 14256, 15400, 15925, 16709, 16896

Offset: 1

Rafik Khalfi, May 30 2024

nonn

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

Cf. A008472 (sopf), A321944.

f := proc (n)
    add(d, d = numtheory[factorset](n))
end proc:
h := proc (n)
    option remember;
    if isprime(n) then
        1
    else
        1+h(convert(numtheory[factorset](n), `+`))    end if:
end proc:
checkDivisibility := proc (n)
    local k, fk, fk1, result:
    result := true:
    fk := n;
    for k from 0 to h(n)-1 do
        fk1 := f(fk);
        if fk1 = 0 or `mod`(fk, fk1) <> 0 then
            result := false:
            break:
        end if:
        fk := fk1:
    end do:
    return result:
end proc:
g := proc (n)
    nops(numtheory[factorset](n)):
end proc:
findNumbers := proc (upper_limit)
    local n, results:
    results := []:
    for n from 2 to upper_limit do
        if checkDivisibility(n) and 2 <= g(n) then
            results := [op(results), n]:
        end if:
    end do:
    return results:
end proc:
upper_limit := 10000:
numbers := findNumbers(upper_limit);

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 *)

A361685 Number of iterations of sopf until reaching a prime.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 3, 1, 2, 0, 2, 0, 1, 3, 1, 2, 1, 0, 3, 2, 1, 0, 2, 0, 1, 2, 2, 0, 1, 1, 1, 2, 3, 0, 1, 2, 2, 2, 1, 0, 2, 0, 4, 2, 1, 2, 2, 0, 1, 4, 3, 0, 1, 0, 3, 2, 3, 2, 2, 0, 1, 1, 1, 0, 2, 2, 3, 2, 1, 0, 2, 2, 2, 2, 2, 2, 1, 0, 2, 3, 1, 0

Offset: 2

J. W. Montgomery, Mar 29 2023

nonn

			a(15) = 2 because 15 is not prime, sopf(15) = 8 is not prime, and sopf^2(15) = sopf(8) = 2 is prime.
a(16) = 1 because 16 is not prime and sopf(16) = 2 is prime.
a(17) = 0 because 17 is prime.

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A008472 (sopf), A321944.

for n=2:101
    s = n;
    c = 0;
    while ~isprime(s)
        s = sum(unique(factor(s)));
        c = c + 1;
    end
    a(n) = c;
end

A008472(n) = vecsum(factor(n)[, 1]);
A361685(n) = for(k=0,oo,if(isprime(n),return(k)); n = A008472(n)); \\ Antti Karttunen, Jan 28 2025

For n >= 2, a(n) = min{m : sopf^m(n) is prime} where sopf^m indicates m iterations of sopf, the sum of the prime factors function.

a(n) = A321944(n) - 1. - Rémy Sigrist, Mar 29 2023

cp's OEIS Frontend

A373285 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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A361685 Number of iterations of sopf until reaching a prime.

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