A281424 Numbers k such that 16*(10^k - 1)/3 + 1 is prime.

6, 23, 65, 82, 108, 188, 300, 342, 401, 584, 1570, 4119, 10030, 24870, 34710

Offset: 1

Author

Sergey Pavlov, Jan 21 2017

Keywords

• nonn
• more

Comments

All prime numbers of the form 16*(10^k - 1)/3 + 1 are terms of A002476.

For any k = a(n), the m-index of 16*(10^k - 1)/3 + 1 in sequence 6m+1 contains exactly a(n) digits, and each digit is 8. E.g., while k = a(1) = 6, 16*(10^6 - 1)/3 + 1 = 6*888888 + 1 = 5333329.

In any number of form 16*(10^k - 1)/3 + 1, its first digit is 5, its two last digits are 29, and each other digit that is between (5...29) is 3.

For k=1, k=2, k=3, the numbers of form 16*(10^k - 1)/3 + 1 are squares of the primes 7, 23, and 73, respectively (see A001248).

Equivalently defined as primes of the form (16*10^k-13)/3. - Tyler Busby, Apr 16 2024

Examples

			For k = a(1) = 6, 16*(10^6 - 1)/3 + 1 = 5333329 and 16*(10^6 - 1)/3 + 1 is prime.
		

Links

• Makoto Kamada, Search for 53w29.

Crossrefs

Cf. A002476.

Programs

Mathematica

Select[Range@ 3000, PrimeQ[16 (10^# - 1)/3 + 1] &] (* Michael De Vlieger, Jan 23 2017 *)

Python

from sympy import isprime
def afind(limit, startk=1):
    pow10 = 10**startk
    for k in range(startk, limit+1):
        if isprime(16*(pow10 - 1)//3 + 1): print(k, end=", ")
        pow10 *= 10
afind(600) # Michael S. Branicky, Aug 17 2021

Extensions

a(12) from Michael S. Branicky, Aug 17 2021

a(13)-a(14) from Michael S. Branicky, Apr 06 2023

a(15) from Kamada data by Tyler Busby, Apr 16 2024