A253295 Prime factor look-and-say sequence starting with a(0) = 8.

8, 32, 52, 22113, 5317113, 131167110613, 1711111711229181533, 1761140131560305063481, 1313718313871371493773936301, 125111501315199577167049112574051, 33185242436199338915435977096119517, 149731486009055371137303679066123116017

Offset: 0

Author

Robert Israel, Dec 29 2014

Keywords

• nonn
• base

Comments

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.

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

Examples

			a(0) = 2^3 so a(1) = 32.
a(1) = 2^5 so a(2) = 52.
a(2) = 2^2 * 13^1 so a(3) = 22113.
a(3) = 3^5 * 7^1 * 13^1 so a(4) = 5317113.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..20
• Mathematics Stack Exchange, Does my "Prime Factor Look and Say" sequence always end?

Crossrefs

Cf. A123132

Programs

Maple

ncat:= (x,y) -> 10^(1+ilog10(y))*x + y:
f:= proc(x) local L,y,t;
  L:= sort(ifactors(x)[2],(a,b)->a[1]
				

Mathematica

a253295[n_] := Block[{a, t = Table[8, {n}]},
  a[x_] := FromDigits[Flatten[IntegerDigits[Reverse /@
  FactorInteger[x]]]]; Do[t[[i]] = a[t[[i - 1]]], {i, 2, n}]; t];
a253295[13] (* Michael De Vlieger, Dec 29 2014 *)

Python

from sympy import factorint
A253295_list = [8]
for _ in range(10):
    A253295_list.append(int(''.join((str(e)+str(p) for p, e in sorted(factorint(A253295_list[-1]).items())))))
# Chai Wah Wu, Dec 30 2014

Formula

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