A263299 Primes that are the concatenation of k 1's, the digits of k^2 + k + 1, and k 1's.

131, 11113111, 1111211111, 111113111111, 11111143111111, 11111111111111111111111

Offset: 1

Author

Altug Alkan, Oct 13 2015

Keywords

• nonn
• base

Comments

Inspiration was a(6) that is concatenation of 10 1's, 10^2 + 10 + 1 and 10 1's. a(6) is R_23 and A004022(3).

k=1, 3, 4, 5, 6, 10 are initial values that generate primes in sequence. The consecutive central polygonal numbers associated with the four consecutive k are 13, 21, 31 and 43.

Note that the middle term of a(2) is 13, not 3.

Next term is too large to include.

The next term has 513 digits. - Harvey P. Dale, Jan 27 2019

Examples

			131 is in the list because 131 is a concatenation of 1, (1^2 + 1 + 1) = 3 and 1, and because 131 is prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..8

Crossrefs

Cf. A002061, A002275, A004022, A068817, A070220, A070746, A261364, A262399.

Programs

Mathematica

Select[FromDigits/@Table[Join[PadRight[{},n,1],IntegerDigits[n^2+n+1],PadRight[{},n,1]],{n,20}],PrimeQ] (* Harvey P. Dale, Jan 27 2019 *)

PARI

for(n=1, 1e3, if(isprime(k=eval(Str((10^n - 1)/9, n^2 + n + 1, (10^n - 1)/9))), print1(k", ")))

Python

from gmpy2 import is_prime
A263299_list = [n for n in (int('1'*k+str(k*(k+1)+1)+'1'*k) for k in range(10**2)) if is_prime(n)] # Chai Wah Wu, Oct 19 2015