A212921 Composite number n = Product(p_j^k_j) that under the iteration of the map Product (p_j^k_j) -> Sum (p_j * k_j) reaches a limit that divides the number itself.

4, 15, 20, 21, 35, 42, 55, 65, 70, 95, 100, 105, 110, 120, 125, 130, 135, 140, 150, 160, 161, 170, 180, 182, 187, 190, 200, 203, 217, 220, 225, 231, 240, 260, 270, 280, 285, 301, 305, 312, 315, 319, 322, 340, 343, 351, 365, 370, 371, 375, 395, 400, 406, 407

Offset

1,1

Comments

Apart from the case n=4, the limit of the iteration is a prime number.

Links

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Example

70 = 2*5*7 -> 2+5+7 = 14 =2*7 -> 2+7=9 = 3^2 -> 3*2=6=2*3 -> 2+3=5 and 70/5=14.

Maple

with(numtheory);
A212921:=proc(q)
local a, b, c, d, i, k, n;
print(4);
for n from 5 to q do
  if not isprime(n) then a:=n;
    while not isprime(a) do
    b:=ifactors(a)[2]; c:=nops(b); b:=op(b); d:=0;
    if c=1 then d:=b[1]*b[2];
    else for k from 1 to c do d:=d+b[k][1]*b[k][2]; od; fi;
    a:=d; if isprime(d) then if trunc(n/d)=n/d then lprint(n, d); fi; break; fi; od;
  fi;
od;
end:
A212921(10000);

Mathematica

it[n_] := it[n] = Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; Dot[p, e]]; it2[n_] := FixedPointList[it[#] &, n]; Select[Range[2, 1000], ! PrimeQ[#] && Mod[#, it2[#][[-1]]] == 0 &] (* T. D. Noe, Jun 01 2012 *)

Crossrefs

Cf. A029909.

Sequence in context: A190709 A166732 A022133 * A359958 A225562 A301707

Adjacent sequences: A212918 A212919 A212920 * A212922 A212923 A212924

Keyword

nonn

Author

Paolo P. Lava, May 31 2012

Status

approved