A250311 -id:A250311 - cp's OEIS frontend

A250312 Numbers which produce primes if their divisors, one by one, are prepended, inserted or appended.

1, 151, 157, 169, 223, 277, 283, 337, 361, 367, 397, 409, 421, 439, 457, 469, 547, 571, 577, 589, 607, 643, 673, 709, 757, 769, 793, 871, 877, 937, 1063, 1093, 1201, 1603, 1609, 1807, 2029, 2053, 2071, 2707, 3019, 3037, 3049, 3073, 3109, 3607, 4039, 4051, 4087

Offset: 1

Paolo P. Lava, Nov 18 2014

			Divisors of 1 is 1 and concat(1,1) = 11 is prime.
Divisors of 151 are 1, 151. Then concat(151,1) = 1511 is prime, as is concat(1,151) = 1151, and concat(1,151,51) = 115151 is prime.
Divisors of 169 are 1, 13, 169. Then concat(16,1,9) = 1619 is prime, concat(16,13,9) = 16139 is prime, as is concat(1,13,69) = 11369, and concat(1,169,69) = 116969 is prime.

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

CF. A250311.

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:=divisors(n); h:=0;
for k from 1 to nops(a) do b:=ilog10(a[k])+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]; else g:=a[k]*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);

Inserted a(3), a(16) and a(26) by Paolo P. Lava, Nov 21 2014

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

A250312 Numbers which produce primes if their divisors, one by one, are prepended, inserted or appended.

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