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Prime factor look-and-say sequence starting with a(0) = 8.
1

%I #20 Jan 02 2015 22:50:58

%S 8,32,52,22113,5317113,131167110613,1711111711229181533,

%T 1761140131560305063481,1313718313871371493773936301,

%U 125111501315199577167049112574051,33185242436199338915435977096119517,149731486009055371137303679066123116017

%N Prime factor look-and-say sequence starting with a(0) = 8.

%C If prime factorization of a(n) is p_1^d_1 p_2^d_2 ... p_k^d_k with p_1 < ... < p_k, then a(n+1) is the concatenation of d_1, p_1, d_2, p_2, ..., d_k, p_k.

%C I suspect that eventually a prime a(n) may be reached, but haven't found one yet.

%H Robert Israel, <a href="/A253295/b253295.txt">Table of n, a(n) for n = 0..20</a>

%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/1084853/does-my-prime-factor-look-and-say-sequence-always-end/1084908#1084908">Does my "Prime Factor Look and Say" sequence always end?</a>

%F a(n+1) = A123132(a(n)).

%e a(0) = 2^3 so a(1) = 32.

%e a(1) = 2^5 so a(2) = 52.

%e a(2) = 2^2 * 13^1 so a(3) = 22113.

%e a(3) = 3^5 * 7^1 * 13^1 so a(4) = 5317113.

%p ncat:= (x,y) -> 10^(1+ilog10(y))*x + y:

%p f:= proc(x) local L,y,t;

%p L:= sort(ifactors(x)[2],(a,b)->a[1]<b[1]);

%p y:= 0;

%p for t in L do y := ncat(y, ncat(t[2],t[1])) od:

%p y

%p end proc:

%p A[0]:= 8:

%p y:= A[0]:

%p for m from 1 to 20 do

%p y:= f(y);

%p A[m]:= y;

%p od:

%p seq(A[i],i=0..20);

%t a253295[n_] := Block[{a, t = Table[8, {n}]},

%t a[x_] := FromDigits[Flatten[IntegerDigits[Reverse /@

%t FactorInteger[x]]]]; Do[t[[i]] = a[t[[i - 1]]], {i, 2, n}]; t];

%t a253295[13] (* _Michael De Vlieger_, Dec 29 2014 *)

%o (Python)

%o from sympy import factorint

%o A253295_list = [8]

%o for _ in range(10):

%o ....A253295_list.append(int(''.join((str(e)+str(p) for p, e in sorted(factorint(A253295_list[-1]).items())))))

%o # _Chai Wah Wu_, Dec 30 2014

%Y Cf. A123132

%K nonn,base

%O 0,1

%A _Robert Israel_, Dec 29 2014