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Numbers k such that (9^k + 8^k)/17 is prime.
+10
15
3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897
COMMENTS
All terms are prime.
a(12) > 10^5.
MATHEMATICA
Select[ Prime[ Range[1, 100000] ], PrimeQ[ (9^# + 8^#)/17 ]& ]
Numbers n such that (10^n + 9^n)/19 is prime.
+10
13
7, 67, 73, 1091, 1483, 10937
COMMENTS
The numbers n themselves (7, 67, 73, ...) are also prime.
a(7) > 10^5.
MATHEMATICA
Select[ Prime[ Range[1, 100000] ], PrimeQ[ (10^# + 9^#)/19 ]& ]
Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.
+10
3
3, 3, 3, 5, 3, 3, 7, 3, 5, 5, 11, 3, 19, 11, 3, 229, 47, 5, 257, 3, 19, 31, 17, 11, 13, 3, 3, 5, 5, 59, 23, 3, 3, 7, 79, 3, 3373, 3, 3, 7, 13, 7, 7, 3527, 593, 19, 3, 3, 13, 13, 11, 19, 41, 3, 7, 109, 3, 227, 13, 5, 5, 3, 239, 5, 3251, 3, 1237, 3, 7, 31, 3, 7
COMMENTS
All terms are odd primes.
a(38),...,a(43) = {3,3,7,13,7,7}.
a(46),...,a(64) = {19,3,3,13,13,11,19,41,3,7,109,11,227,13,5,5,3,239,5}.
a(66) = 3. a(68),...,a(72) = {3,7,31,3,7}.
a(74),...,a(92) = {3,5,19,17,3,83,3,3,19,19,11,11,61,3,7,7,3,41,29}.
a(94) = 5. a(97),a(98) = {19,7}. a(100) = 31.
a(n) is currently unknown for n = {37,44,45,65,67,73,93,95,96,99,...}.
All known terms from n=1..100 correspond to proven primes.
a(96) > 10250.
a(99) > 10250. (End)
Presuming every prime is seen at least once, one can specifically seek those with fixed k. Doing this, a(174) = 37, a(368) = 43 for example. - Bill McEachen, Aug 26 2024
EXAMPLE
For n=4, the expression (2^k + (2n-1)^k)/(2n+1) takes on values 1, 53/9, 39, 2417/9, and 1871 for k=1..5. Since 1871 is the first prime number to occur, a(4) = 5.
MATHEMATICA
Do[k = 1; While[ !PrimeQ[(2^k + (2n-1)^k)/(2n+1)], k++ ]; Print[k], {n, 100}] (* Ryan Propper, Mar 29 2007 *)
CROSSREFS
Cf. A082387 ((2^n + 5^n)/7 is prime), A125955 ((2^n + 7^n)/9 is prime). Cf. A125956 ((2^n + 9^n)/11 is prime), A125955 ((2^n + 11^n)/13 is prime).
Numbers n such that (15^n + 2^n)/17 is prime.
+10
3
3, 67, 199, 479, 563, 2243, 2579, 6599, 7951, 10099, 10909, 13759
COMMENTS
All terms are odd primes.
a(13) > 10^5.
MATHEMATICA
Select[Prime[Range[1, 100000]], PrimeQ[(15^# + 2^#)/17]&]
PROG
(PARI) forprime(p=3, 10^6, if(ispseudoprime((15^p + 2^p)/17), print1(p, ", ") ) ); \\ Joerg Arndt, Jul 29 2013
EXTENSIONS
Removed incorrect first term of "2".
Numbers n such that (13^n + 2^n)/15 is prime.
+10
2
7, 31, 103, 223, 503, 1171, 1973, 4111, 4729
COMMENTS
All terms are prime.
a(10) > 10^5.
MATHEMATICA
Select[Prime[Range[1, 100000]], PrimeQ[(13^# + 2^#)/15]&]
PROG
(PARI) forprime(p=3, 10^6, if(ispseudoprime((13^p + 2^p)/15), print1(p, ", ") ) ); \\ Joerg Arndt, Jul 29 2013
EXTENSIONS
Removed incorrect first term of "2".
Numbers n such that (9^n + 4^n)/13 is prime.
+10
1
3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251
COMMENTS
All terms are prime.
The next element, a(19), is greater than 10^5.
MATHEMATICA
Select[ Prime[ Range[1, 100000] ], PrimeQ[ (9^# + 4^#)/13 ]& ]
Numbers k such that (9^k + 7^k)/16 is prime.
+10
0
3, 107, 197, 2843, 3571, 4451, 31517, 44819
COMMENTS
All terms are prime.
The corresponding primes are 67, 79401467172644850007356716446663549450843749853576087044440771380676673442288169290888310265443988907, ...
MAPLE
select(n->isprime((9^n+7^n)/16), [seq(n, n=1..10000, 2)]); # Muniru A Asiru, Mar 27 2018
MATHEMATICA
Select[Range[1, 10000], PrimeQ[(9^n+7^n)/16] &]
PROG
(PARI) forprime(n=3, 10000, if(isprime((9^n+7^n)/16), print1(n, ", ")))
(Magma) [n: n in [1..10000] |IsPrime((9^n+7^n)/16)]
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