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Numbers k such that 4^k + 13 is prime.
+10
8
1, 2, 4, 10, 19, 32, 40, 146, 566, 2054, 9967, 62639, 87814, 141092
COMMENTS
Numbers of the form 4^n+k (for n>0) are never primes when k is even (obviously) or when k == -1 (mod 6): in the last case, in fact, (3+1)^n + 6*h-1 is divisible by 3. - Bruno Berselli, Oct 06 2015
MATHEMATICA
Select[Range[4000], PrimeQ[4^# + 13] &]
PROG
(Magma) [n: n in [0..2000] | IsPrime(4^n+13)];
CROSSREFS
Cf. Numbers k such that 4^k + d is prime: A089437 (d=3), A217349 (d=7), A217350 (d=9), this sequence (d=13), A253773 (d=15), A253774 (d=19), A262345 (d=21), A204388 (d=25), A262969 (d=27), A262971 (d=31), A262972 (d=33).
Numbers n such that 8^n + 3 is prime.
+10
6
1, 2, 4, 5, 6, 10, 28, 76, 130, 370, 568, 713, 789, 790, 1334, 1354, 1849, 2913, 5729, 5740, 5978, 6908, 10618, 11918, 12748, 13449, 40850, 68654, 78442, 121040, 159948, 228526
MATHEMATICA
Select[Range[5000], PrimeQ[8^# + 3] &]
Primes of the form 4^k + 3.
+10
5
7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, 19342813113834066795298819
EXAMPLE
67 is a term because 4^3 + 3 = 67 is prime.
MATHEMATICA
Select[Table[4^n + 3, {n, 0, 200}], PrimeQ]
PROG
(Magma) [a: n in [0..200] | IsPrime(a) where a is 4^n+3];
CROSSREFS
Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), this sequence (r=4, h=3), A228027 (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).
Numbers k such that 4^k + 7 is prime.
+10
3
1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 19, 22, 39, 44, 49, 63, 80, 87, 102, 107, 294, 305, 399, 463, 595, 599, 903, 944, 1324, 1727, 1755, 1932, 1935, 4485, 6165, 6665, 9438, 11169, 19859, 27503, 55392, 86235, 98217, 117855, 123640, 134204, 139660, 150437, 157634, 186475, 236129, 283248, 390142, 410178
EXAMPLE
For k = 14, 4^14 + 7 = 268435463 is prime.
MATHEMATICA
Select[Range[0, 5000], PrimeQ[4^# + 7] &]
Numbers k such that 4^k + 9 is prime.
+10
2
1, 3, 5, 9, 15, 33, 41, 335, 443, 671, 1197, 1355, 2247, 2639, 117293, 191099
COMMENTS
Contains exactly the halved even terms of A057196.
EXAMPLE
For k = 15, 4^15 + 9 = 1073741833 is prime.
MATHEMATICA
Select[Range[0, 5000], PrimeQ[4^# + 9] &]
Smallest k >= 1 such that (n-1)*n^k + 1 is prime.
+10
1
1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1
COMMENTS
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
PROG
(PARI) a(n)=for(k=1, 2^16, if(ispseudoprime((n-1)*n^k+1), return(k)))
CROSSREFS
For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
b a1(b) a2(b) a3(b) a4(b) a5(b) a6(b) a7(b) a8(b)
--------------------------------------------------------------------
13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx 14 A273523 ------- ------- ------- ------- ------- ------- ------- 15 ------- ------- ------- ------- ------- ------- ------- -------
16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
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