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A331867
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Numbers n for which R(n) + 3*10^floor(n/2-1) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).
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3
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OFFSET
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1,1
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COMMENTS
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The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 3*10^floor(n/2-1) has ceiling(n/2) digits 1, one digit 4, and again floor(n/2-1) digits 1. For odd and even n, the digit 4 is just to the right of the middle of the number.
For odd n = 2m + 1, f(n) = R(n) + 3*10^floor(n/2-1) is divisible by 3, 7 or 13 when m is congruent 1 or 4, 3 or 5, resp. 0 or 2 (mod 6): there can't be an odd term.
For even n = 2m, f(n) is divisible by 3 or 7 when m is congruent to 0 or 3, resp. 1 or 2 (mod 6). When m = 6k + 4, then f(n) is prime for k = 5 and 437 (and no further k <= 600), and divisible by 23 or 53 when k is congruent to 10 (mod 11) resp. 3 (mod 13). When m = 6k + 5, f(n) is prime for k = 457 and 591 and no other value up to 600, and divisible by 23, 47, 53, 97, 163, 181, 859, ... for k congruent to 5 (mod 11), 11 (mod 23), 5 (mod 13), 0 (mod 32), 13 (mod 27), 26 (mod 30), 3 (mod 13), ..., respectively.
a(5) > 7272.
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LINKS
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EXAMPLE
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For n = 2, R(2) + 3*10^floor(2/2-1) = 14 = 2*7 is not prime.
For n = 3, R(3) + 3*10^floor(3/2-1) = 114 = 2*3*19 is not prime.
For n = 4, R(4) + 3*10^floor(4/2) = 1141 = 7*163 is not prime.
For n = 5, R(5) + 3*10^floor(5/2) = 11141 = 13*857 is not prime.
For n = 68, R(68) + 3*10^floor(68/2) = 1...1141...1 is prime, with 34 digits '1' to the left of a digit '4' and 33 digits '1' to its right.
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MATHEMATICA
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Select[Range[2, 2500], PrimeQ[(10^# - 1)/9 + 3*10^Floor[#/2 - 1]] &] (* corrected by Amiram Eldar, Feb 10 2020 *)
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PROG
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(PARI) for(n=2, 9999, isprime(p=10^n\9+3*10^(n\2-1))&&print1(n", "))
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CROSSREFS
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KEYWORD
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nonn,base,hard,more
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AUTHOR
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STATUS
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approved
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