|
%I #52 Jan 18 2024 02:54:22
%S 53,757,2493,3597,5835,46069,95019,104281,134809
%N Odd integers k such that 10^k - 1 - 10^((k-1)/2) is a prime of the form 9...989...9, called a palindromic wing prime or a near-repdigit palindromic prime.
%C Prime versus probable prime status and proofs are given in the author's table.
%C The corresponding primes have a(n) digits all of which are '9's except for the middle digit which is an '8'. They are too large to be listed in a sequence on their own, cf. examples. See A077775-A077798 and A107123-A107127 for palindromic wing/near-repdigit primes with other digits. - _M. F. Hasler_, Mar 03 2019
%C 1888529 is a term but its position is not known. - _Jeppe Stig Nielsen_, Jan 12 2024
%C a(10) > 600000. - _Serge Batalov_, Jan 17 2024
%D C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.
%H Patrick De Geest, World!Of Numbers, <a href="http://www.worldofnumbers.com/wing.htm#pwp989">Palindromic Wing Primes (PWP's)</a>
%H Makoto Kamada, <a href="https://stdkmd.net/nrr/9/99899.htm#prime">Prime numbers of the form 99...99899...99</a>
%H PrimePages, <a href="https://t5k.org/primes/search.php?Description=10%5E%25-10%5E%25-1&Comment=palindrome%25near-repdigit+OR+near-repdigit%25palindrome&OnList=all&Style=HTML">Database Search Output</a>.
%H <a href="/index/Pri#Pri_rep">Index entries for primes involving repunits</a>.
%F a(n) = 2*A183187(n) + 1.
%e a(1) = 53 corresponds to the 53-digit prime
%e p = 99999999999999999999999999899999999999999999999999999.
%e a(2) = 757 corresponds to p = (10^757 - 1) - 10^378 = 99...99899...99.
%t Do[ If[ PrimeQ[10^n - 1*10^Floor[n/2] - 1], Print[n]], {n, 3, 104300, 2}] (* _Robert G. Wilson v_, Dec 16 2005 *)
%o (PARI) is(n)=bittest(n,0)&&ispseudoprime(10^n-1-10^(n\2))
%o forstep(n=1,oo,2,is(n)&&print1(n",")) \\ _M. F. Hasler_, Mar 03 2019
%Y Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.
%K more,nonn,base
%O 1,1
%A _Patrick De Geest_, Nov 16 2002
%E a(9) from PWP table, added by _Patrick De Geest_, Nov 05 2014
| 