A054416 Numbers k such that 9090...9091 (with k-1 copies of 90 and one copy of 91) is prime.

2, 3, 9, 15, 26, 33, 146, 320, 1068, 1505, 134103, 800393

Offset: 1

Author

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 22 2000

Keywords

• nonn
• more
• hard

Comments

Numbers k such that 10*(10^(2k)-1)/11 + 1 is prime.

Examples

			The first 3 numbers are 9091, 909091, 909090909090909091.
		

References

• J. A. H. Hunter and J. S. Madachy, Mathematical Diversions, New York: Dover Publications, Inc., 1974, pp. 4-5. Originally published by Van Nostrand, 1963.

Links

• David Broadhurst, Proof that 1505 term is prime [BROKEN LINK]
• David Broadhurst, Unique record, digest of 3 messages in primenumbers Yahoo group, Apr 8-9, 2001. [Cached copy]

Crossrefs

Cf. A001562.

Programs

Mathematica

Do[ If[ PrimeQ[ 10*(10^(2n) - 1)/11 + 1], Print[ n ] ], {n, 0, 1505} ]
Position[Table[FromDigits[PadLeft[{9,1},2n,{9,0}]],{n,1510}], ?PrimeQ]// Flatten (* _Harvey P. Dale, Nov 02 2017 *)

Python

from sympy import isprime, prime
def afind(limit, startk=1):
    s = "90"*(startk-1)
    for k in range(startk, limit+1):
        if isprime(int(s+"91")):
            print(k, end=", ")
        s += "90"
afind(400) # Michael S. Branicky, Jan 13 2022

Formula

a(n) = (A001562(n)-1)/2.

Extensions

More terms from Michael Kleber and Harvey Dubner (harvey(AT)dubner.com), May 22 2000

Ignacio Larrosa Cañestro reports that the 1068 term has now been established to be a prime using Titanix 1.01, Oct 23 2000

a(11)-a(12) from Michael S. Branicky, Jan 13 2022 using A001562