login
A107648
Numbers n such that (10^(2n+1)+63*10^n-1)/9 is prime.
45
1, 4, 6, 7, 384, 666, 675, 3165, 131020
OFFSET
1,2
COMMENTS
n is in the sequence iff the palindromic number 1(n).8.1(n) is prime (dot between numbers means concatenation). Let f(n)=(10^(2n+1)+63*10^n-1)/9 then for all nonnegative integers m we have: I. 3 divides f(3m+2) II. 19 divides f(18m+13) III. 29 divides f(28*m+16) & 29 divides f(28*m+25) IV. 31 divides f(30*m+2) & 31 divides f(30*m+17) V. 41 divides f(5m+3), etc. So if n is in the sequence then n is not of the forms 3m+2, 18m+13, 28m+16 28m+25, 30m+2, 30m+17, 5m+3, etc.
a(9) > 10^5. - Robert Price, Oct 30 2017
REFERENCES
C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 141.
FORMULA
a(n) = (A077791(n)-1)/2.
EXAMPLE
7 is in the sequence because (10^15+63*10^7-1)/9=1(7).8.1(7)=111111181111111 is prime.
666 is in the sequence because (10^(2*666+1)+63*10^666-1)/9=1(666).8.1(666) is prime.
MATHEMATICA
Do[If[PrimeQ[(10^(2n + 1) + 63*10^n - 1)/9], Print[n]], {n, 4000}]
PROG
(PARI) for(n=0, 1e4, if(ispseudoprime(t=(10^(2*n+1)+63*10^n)\9), print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011
KEYWORD
nonn,more,base
AUTHOR
Farideh Firoozbakht, May 19 2005
EXTENSIONS
Edited by Ray Chandler, Dec 28 2010
a(9) from Robert Price, Aug 03 2024
STATUS
approved