1,2

Comments from Walter Kehowski (wkehowski(AT)cox.net), Jun 28 2006: "The sequence of primes 2*6^n-1 is best looked at in base 12, with X for 10 and E for 11. The first few powers of 6 are 1, 6, 30, 160, .. and so all powers of 6 after the first are divisible by 12. Recall that in base 12 all primes>3 end in the digits 1, 5, 7, E. Thus all terms of the sequence 2*6^n-1 are E mod 12 and so all primes are E-primes.

"The first few terms of 2*6^n-1 in base 12 are n: E, 5E, 2EE, 15EE, 8EEE, X15EEEEEE, 7715EEEEEEEE, 1099345EEEEEEEEEEEEE, 64X782EEEEEEEEEEEEEE, 1975XE415EEEEEEEEEEEEEEEEEE, 1427526115EEEEEEEEEEEEEEEEEEEE, 60E9553508EEEEEEEEEEEEEEEEEEEEEEE, 468X10E6968EEEEEEEEEEEEEEEEEEEEEEEEE, 6X1317542415EEEEEEEEEEEEEEEEEEEEEEEEEE, 350769881208EEEEEEEEEEEEEEEEEEEEEEEEEEE, 2528640575X776182EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE, and so on. The last known element 2*6^33003-1 ends in hundreds of E's.

"Similar conclusions can be drawn about similar sequences. For example, we have the following possibilities, sometimes only after the first term: A000043, A000668: n+1 prime, 2*2^n-1, 7-primes; A003307, A079363: n, 2*3^n-1, 5-primes; JS, JS: n even, 2*5^n-1, 1-primes; A002959, NYS: n, 2*7^n-1, 1-primes; A000043, A000668: 3n+1 prime, 2*8^n-1, 7-primes; A003307, A079363: 2n, 2*9^n-1, 5-primes; A002957, A055558: n, 2*X^n-1, 7-primes; JS, JS: 2*E^n-1, n even, 1-primes. JS means "Just submitted" (must replace by appropriate A-numbers)."

Do[ If[ primeQ[ 2*6^n - 1], Print[n]], {n, 1, 1500}]

Sequence in context: A132027 A103651 A093713 this_sequence A117577 A069469 A109849

Adjacent sequences: A057469 A057470 A057471 this_sequence A057473 A057474 A057475

nonn

Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 10 2000

More terms from Pierre CAMI (pierrecami(AT)tele2.fr), Jun 16 2006