A210513 - OEIS

%I #45 Jul 26 2022 21:58:52

%S 227,337,557,887,997,11117,24247,26267,27277,29297,30307,32327,39397,

%T 48487,51517,54547,60607,62627,65657,68687,69697,72727,74747,78787,

%U 81817,87877,89897,90907,92927,93937,95957,101710177,101910197,103110317,103410347,103810387

%N Primes formed by concatenating k, k, and 7.

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

%C Base considered is 10.

%C Observations:

%C - k cannot be a multiple of 7.

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

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

%C - 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.

%H Michael S. Branicky, <a href="/A210513/b210513.txt">Table of n, a(n) for n = 1..10000</a>

%e For k = 2, a(1) = 227.

%e For k = 3, a(2) = 337.

%e For k = 5, a(3) = 557.

%e For k = 8, a(4) = 887.

%e For k = 9, a(5) = 997.

%t Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], {7}}]], {n, 100}], PrimeQ] (* _Alonso del Arte_, Feb 01 2013 *)

%o (Python)

%o import numpy as np

%o from functools import reduce

%o def factors(n):

%o     return reduce(list.__add__, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))

%o for i in range(1, 2000):

%o     p1=int(str(i)+str(i)+"7")

%o     if len(factors(p1))<3:

%o         print(p1, end=',')

%o (Python)

%o from sympy import isprime

%o from itertools import count, islice

%o def agen(): yield from filter(isprime, (int(str(k)+str(k)+'7') for k in count(1)))

%o print(list(islice(agen(), 36))) # _Michael S. Branicky_, Jul 26 2022

%Y Cf. A030458, A052089, A092994.

%K base,nonn,easy

%O 1,1

%A _Abhiram R Devesh_, Jan 26 2013

%E a(34) and beyond from _Michael S. Branicky_, Jul 26 2022