A250312 -id:A250312 - cp's OEIS frontend

A250311 Numbers which produce primes if their prime factors, one by one, are prepended, inserted or appended.

49, 131, 133, 149, 151, 157, 169, 173, 179, 191, 197, 199, 223, 233, 239, 247, 277, 281, 283, 293, 313, 331, 337, 361, 367, 383, 397, 401, 409, 419, 421, 431, 439, 443, 457, 463, 467, 469, 481, 503, 547, 553, 571, 577, 587, 589, 607, 641, 643, 659, 673, 679, 701

Offset: 1

Paolo P. Lava, Nov 18 2014

			Prime factors of a(1) = 49 are 7, 7 and concat(4,7,9) = 479 is prime.
a(2) = 131 is prime and concat(13,131,1) = 131311 is prime, as is concat(1,131,31) = 113131.
Prime factors of a(3) = 14383 are 19, 757. Then, concat(1,19,4383) = 1194383 is prime and concat(1438,757,3) = is prime, as is concat(14,757,383) = 14757383.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

CF. A250312.

with(numtheory): P:=proc(q) local a,b,c,f,g,h,j,k,n;
for n from 1 by 2 to q do a:=ifactors(n)[2]; h:=0;
for k from 1 to nops(a) do b:=ilog10(a[k][1])+1;
for j from 0 to ilog10(n)+1 do f:=(n mod 10^j);
if j=0 then c:=n*10^b+a[k][1]; else g:=a[k][1]*10^(ilog10(f)+1)+f;
c:=trunc(n/10^j)*10^(ilog10(g)+1)+g; fi;
if isprime(c) then h:=h+1; break; fi; od;
if h=nops(a) then print(n); fi; od; od; end: P(10^6);

A247098 Smallest prime p that generates n different primes if it is inserted into p itself.

Original entry on oeis.org

2, 109, 131, 10289, 12197, 1227797, 101636629, 118561139, 10596073217, 89466662147, 37898818253

Offset: 0

Paolo P. Lava, Nov 18 2014

a(8) > 10^9.

The concatenation of p with p will not result in a prime as the result is divisible by 10^k+1, where k is the number of digits in p. - Chai Wah Wu, Jun 25 2019

			a(0) = 2 is prime but concat(2,2) = 22 is not.
a(1) = 109 is prime and also concat(1,109,09) = 110909.
a(2) = 131 is prime. Again, concat(13,131,1) = 131311 is prime and also concat(1,131,31) = 113131.
a(5) = 1227797 and the five primes that are generated are: 12277912277977, 12271227797797, 12212277977797, 12122779727797, 11227797227797.

Cf. A250311, A250312.

P:=proc(q) local a,b,i,j,k,n,t: print(2); for j from 1 to q do n:=2:
for k from 1 to q do n:=nextprime(n): b:=length(n): t:=0:
for i from 1 to b-1 do if isprime((trunc(n/10^i)*10^b+n)*10^i+(n mod 10^i))
then t:=t+1; fi; od;  if t=j then print(n); break; fi; od; od; end: P(10^9);

a(8)-a(10) from Chai Wah Wu, Jul 01 2019

cp's OEIS Frontend

A250311 Numbers which produce primes if their prime factors, one by one, are prepended, inserted or appended.

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