A029909 -id:A029909 - cp's OEIS frontend

A146071 Consider A145834 as the first step of the sieving (subtracting the sum of its prime factors with repetition from the composite numbers). This sequence is the result of the subsequent application of above described sieving - thus all terms of this sequence arise as prime numbers.

0, 1, 2, 3, 3, 5, 5, 7, 2, 3, 11, 11, 3, 7, 7, 11, 3, 17, 11, 3, 19, 7, 23, 11, 17, 23, 29, 11, 29, 7, 11, 37, 23, 17, 31, 23, 43, 23, 43, 23, 3, 37, 29, 17, 23, 47, 17, 47, 43, 43, 37, 23, 29, 53, 59, 37, 67, 43, 23, 43, 17, 41, 23, 71, 59, 71, 47, 59, 7, 71, 83, 23, 23, 41, 67, 17, 59

Offset: 1

Alexander R. Povolotsky, Oct 26 2008

nonn

Florentin Smarandache (in a Sunday, Nov 02 2008 email exchange) asks: how many times does each prime (> 3) appear in this sequence? This question can also be asked about A029909. - Alexander R. Povolotsky, Nov 07 2008

Cf. A145834, A001414, A002808, A075255.

A146071(n)=n=A002808(n); while( n>1 & !isprime(n), n=A075255(n)); n \\ M. F. Hasler, Nov 02 2008

More terms from M. F. Hasler, Nov 02 2008

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.

Original entry on oeis.org

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

Paolo P. Lava, May 31 2012

nonn

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

			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.

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

Cf. A029909.

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);

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 *)

cp's OEIS Frontend

A146071 Consider A145834 as the first step of the sieving (subtracting the sum of its prime factors with repetition from the composite numbers). This sequence is the result of the subsequent application of above described sieving - thus all terms of this sequence arise as prime numbers.

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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.

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