A210513 Primes formed by concatenating k, k, and 7.

227, 337, 557, 887, 997, 11117, 24247, 26267, 27277, 29297, 30307, 32327, 39397, 48487, 51517, 54547, 60607, 62627, 65657, 68687, 69697, 72727, 74747, 78787, 81817, 87877, 89897, 90907, 92927, 93937, 95957, 101710177, 101910197, 103110317, 103410347, 103810387

Offset: 1

Author

Abhiram R Devesh, Jan 26 2013

Keywords

• base
• nonn
• easy

Comments

This sequence is similar to A030458, A052089, and A092994.

Base considered is 10.

Observations:

- k cannot be a multiple of 7.

- k cannot have a digital root 7 as the sum of the digits would be divisible by 3.

- There is no k between 100 and 1000 that can form a prime number of this form after 95957 the next prime is 101710177.

- k cannot have a digital root equal to 1 or 4, because then in the concatenation it contributes 2 or 8 to the digital root of the number, and that number is then divisible by 3.

Examples

			For k = 2, a(1) = 227.
For k = 3, a(2) = 337.
For k = 5, a(3) = 557.
For k = 8, a(4) = 887.
For k = 9, a(5) = 997.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A030458, A052089, A092994.

Programs

Mathematica

Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], {7}}]], {n, 100}], PrimeQ] (* Alonso del Arte, Feb 01 2013 *)

Python

import numpy as np
from functools import reduce
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
for i in range(1, 2000):
    p1=int(str(i)+str(i)+"7")
    if len(factors(p1))<3:
        print(p1, end=',')

Python

from sympy import isprime
from itertools import count, islice
def agen(): yield from filter(isprime, (int(str(k)+str(k)+'7') for k in count(1)))
print(list(islice(agen(), 36))) # Michael S. Branicky, Jul 26 2022

Extensions

a(34) and beyond from Michael S. Branicky, Jul 26 2022