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A373285
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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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0
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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For k = 11475 = 3^3 * 5^2 * 17, sopf(k)=25 divides k and sopf(sopf(k))=5 divides sopf(k).
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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