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

Author

Bill McEachen, May 26 2013

Keywords

• easy
• nonn
• base

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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++);
); }