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A210513
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Primes formed by concatenating k, k, and 7.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], {7}}]], {n, 100}], PrimeQ] (* Alonso del Arte, Feb 01 2013 *)
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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base,nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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