A034093 Number of near-repunit primes that can be formed from (10^k - 1)/9 by changing just one digit from 1 to 0.

0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 5, 0, 0, 0, 0, 2, 5, 0, 4, 0, 0, 0, 3, 0, 1, 0, 0, 1, 2, 0, 4, 1, 0, 1, 2, 0, 2, 1, 0, 0, 7, 0, 4, 0, 0, 0, 2, 0, 2, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 1, 3, 0, 1, 0, 0, 1, 3, 0, 3, 0, 0, 1, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 0

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base

Examples

			a(12) = 5 because from (10^12 - 1)/9 = 111111111111, by changing just one digit from 1 to 0, out of the eleven candidates, 111111111101, 111111110111, 111111011111, 111011111111 and 101111111111 are primes.
		

References

• C. K. Caldwell and H. Dubner, The near repunits primes, J. Rec. Math., Vol. 27(1), 1995, pp. 35-41.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..2000
• Chris Caldwell, Below are all of the 12-digit Near-Repunit primes.
• Chris Caldwell, Repunits.

Crossrefs

Cf. A004022, A065074, A065083.

Programs

Mathematica

a = {}; Do[ p = IntegerDigits[ (10^n - 1)/9 ]; c = 0; Do[ If[ q = FromDigits[ ReplacePart[p, 0, i]]; PrimeQ[q], c++ ], {i, 2, n} ]; a = Append[a, c], {n, 1, 100} ]; a (* Robert G. Wilson v, Nov 19 2001 *)

PARI

a(n)=sum(i=1,n-2,ispseudoprime(10^n\9-10^i)) \\ Charles R Greathouse IV, May 01 2012

Python

from sympy import isprime
def a(n):
    Rn = (10**n-1)//9
    return sum(1 for i in range(n-1) if isprime(Rn-10**i))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Nov 04 2023

Extensions

More terms from Robert G. Wilson v, Nov 19 2001

Edited by N. J. A. Sloane, Oct 02 2008 at the suggestion of R. J. Mathar