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

Author

Paolo P. Lava, May 31 2012

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A029909.

Programs

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 *)