A136296 "Special augmented primes": primes p such that the decimal number 1p1 is divisible by p.

11, 13, 137, 9091, 909091, 5882353, 909090909090909091, 909090909090909090909090909091, 9090909090909090909090909090909090909090909090909091, 909090909090909090909090909090909090909090909090909090909090909091

Offset: 1

Author

N. J. A. Sloane, Apr 20 2008

Keywords

• nonn
• base

Comments

Equals A116436 INTERSECT A000040. - R. J. Mathar, Apr 24 2008

The larger terms may be only probable primes. - Franklin T. Adams-Watters, Apr 23 2008

According to the Magma Calculator (http://magma.maths.usyd.edu.au/calc/), all nine terms given for this sequence are prime. - Jon E. Schoenfield, Aug 24 2009

Examples

			11371/137 = 83, an integer, so the prime 137 is a term.
		

References

• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 61.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..12 (all terms with <= 1000 digits)

Crossrefs

Prime members of A116436.

Programs

Mathematica

max=6; a={}; For[i=1, i<=10^max, i++, If[Mod[FromDigits[Join[{1}, IntegerDigits[Prime[i]], {1}]], Prime[i]] == 0, AppendTo[a, Prime[i]]]]; a (* Stefano Spezia, Mar 26 2023 *)

PARI

A136296k(k) = { local(l, d, lb, ub); d=factor(10^(k+1)+1)[,1]; l=[]; lb=10^(k-1); ub=10*lb; for(i=1,#d,if(d[i]>=lb&&d[i]A136296k(k))) \\ Franklin T. Adams-Watters, Apr 23 2008

Python

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for k in count(2):
        t = 10**(k+1) + 1
        d = [t//i for i in range(100, 10, -1) if t%i == 0]
        yield from (di for di in d if isprime(di))
print(list(islice(agen(), 8))) # Michael S. Branicky, Mar 26 2023 following Franklin T. Adams-Watters but removing factorization

Extensions

a(4)-a(6) from M. F. Hasler, Apr 22 2008

a(7)-a(9) from Franklin T. Adams-Watters, Apr 23 2008

a(10) from Michael S. Branicky, Mar 26 2023