A123132 -id:A123132 - cp's OEIS frontend

A226095 Primes formed by concatenation (exponent then prime) of prime factorizations of the positive integers.

13, 1213, 17, 23, 2213, 113, 1217, 1223, 12113, 131, 137, 12119, 22111, 3217, 167, 173, 179, 43, 221317, 12143, 22123, 197, 1103, 1109, 4217, 22129, 17117, 211, 12161, 32117, 13147, 1327, 1151, 32119, 23117, 15131, 17123, 1163, 1213129, 13159, 1181, 1217113

Offset: 1

Bill McEachen, May 26 2013

This produces primes well above the base composite, like 22111 from 44.
Entries stemming strictly from composite prime factorizations will be unique and the sequence will very likely be infinite. Of course not every prime will be encountered, and duplication will be seen across composite and prime factorization treated jointly (an example being 18 and 223 both yielding the prime 1223).
Primes in A123132. - Charles R Greathouse IV, May 28 2013

			44 = 2^2 * 11^1 yields 22111, which is prime and so enters the sequence.  Powers precede the prime factor.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A105435 (primes which with a 1 prepended stay prime).

select(isprime, [seq((l-> parse(cat(seq([i[2], i[1]][], i=l))))(sort(ifactors(n)[2], (x, y)-> x[1]Alois P. Heinz, Nov 24 2017

t = {}; Do[s = FromDigits[Flatten[IntegerDigits /@ RotateLeft /@ FactorInteger[n]]]; If[PrimeQ[s], AppendTo[t, s]], {n, 2, 200}]; t (* T. D. Noe, May 28 2013 *)
Select[FromDigits[Flatten[IntegerDigits/@Reverse/@FactorInteger[#]]]&/@ Range[2, 300],PrimeQ] (* Harvey P. Dale, Nov 24 2017 *)

list(maxx)={
n=3;cnt=0;
while(n<=maxx,
f=factorint(n); old=0;
\\ as we concatenate, code is f{digits of each p.f.&pwr}
for (i=1,#f[,1],
new=(10^length(  Str(f[i,1]) )  *f[i,2] + f[i,1]);
q=new+(10^length(Str(new))   )*old;  old=q  );
if(isprime(q),  print("entry from", n, "   ",  q);
cnt++);  n++;
while(isprime(n),n++);
); }

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

Original entry on oeis.org

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

Offset: 0

Robert Israel, Dec 29 2014

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.

			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.

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

Cf. A123132

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]

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 *)

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

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

cp's OEIS Frontend

A226095 Primes formed by concatenation (exponent then prime) of prime factorizations of the positive integers.

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