A262399 -id:A262399 - cp's OEIS frontend

A261364 Semiprimes that are the concatenation of n 1's, 2^n and n 1's.

121, 1118111, 1111161111, 11111164111111, 111111111512111111111, 1111111111114096111111111111, 111111111111181921111111111111, 111111111111111638411111111111111, 111111111111111111262144111111111111111111, 11111111111111111111104857611111111111111111111

Offset: 1

Altug Alkan, Oct 02 2015

Inspiration was symmetry and visual simplicity.

			a(1) = 121 because the concatenation of 1, 2 and 1 is a semiprime number.
a(2) = 1118111 because the concatenation of 111, 8 and 111 is a semiprime number.
a(3) = 1111161111 because the concatenation of 1111, 16 and 1111 is a semiprime number.

Cf. A001358, A002275, A068817, A070220, A070746, A262399.

ncat:= (a,b) -> 10^(1+ilog10(b))*a+b:
f:= proc(n) local N;
  N:= ncat(ncat((10^n-1)/9,2^n),(10^n-1)/9);
  if numtheory:-bigomega(N) = 2 then N else NULL fi
end proc:
seq(f(n),n=1..25); # Robert Israel, Oct 04 2015

Select[Table[FromDigits[Flatten[{PadRight[{},n,1],IntegerDigits[2^n],PadRight[{},n,1]}]],{n,20}], PrimeOmega[#]==2&] (* Harvey P. Dale, Dec 02 2023 *)

for(n=1, 25, if(bigomega(k=eval(Str((10^n - 1)/9, 2^n, (10^n - 1)/9))) == 2, print1(k", ")))

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

Original entry on oeis.org

131, 11113111, 1111211111, 111113111111, 11111143111111, 11111111111111111111111

Offset: 1

Altug Alkan, Oct 13 2015

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

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

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

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

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

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

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

cp's OEIS Frontend

A261364 Semiprimes that are the concatenation of n 1's, 2^n and n 1's.

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