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A331862
Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.
4
3, 26, 186, 206, 258, 3486, 12602
OFFSET
1,1
COMMENTS
The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.
In base 10, R(n) - 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 0, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 0 is just left to the middle of the number.
There can't be an odd term > 3 because the corresponding palindrome factors as R((n-1)/2)*(10^((n+1)/2) + 1).
No term can be congruent to 1 mod 3. - Chai Wah Wu, Feb 07 2020
LINKS
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015)
EXAMPLE
For n = 3, R(n) - 10^floor(n/2) = 101 is prime.
For n = 26, R(n) - 10^floor(n/2) = 11111111111101111111111111 is prime.
PROG
(PARI) for(n=0, 9999, isprime(p=10^n\9-10^(n\2))&&print1(n", "))
CROSSREFS
Cf. A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A105992 (near-repunit primes), A138148 (Cyclops numbers with digits 0 & 1).
Cf. A331860 (variant with digit 2 instead of digit 0), A331863 (variant with floor(n/2-1) instead of floor(n/2)).
Sequence in context: A038697 A226351 A091262 * A364634 A121121 A355047
KEYWORD
nonn,base,hard,more
AUTHOR
M. F. Hasler, Jan 30 2020
EXTENSIONS
a(6)-a(7) from Giovanni Resta, Jan 31 2020
STATUS
approved