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Numbers k such that 4^k + 25 is prime.
4

%I #37 Nov 28 2023 17:48:20

%S 1,2,3,4,5,10,11,17,35,46,56,59,118,125,189,219,285,327,400,818,1424,

%T 2474,2835,3386,3747,4003,4528,5519,8134,10708,10869,16685,39353,

%U 56065,63223,82023,109625,118216,184024,262077,265405,320427,349870,373151,377019,377188,465988,494781

%N Numbers k such that 4^k + 25 is prime.

%C a(40) > 2.5*10^5. - _Robert Price_, Oct 17 2015

%C Since 4^(6k) + 25 = 4096^k + 25 == (1^k + 25) mod 13 = 26 mod 13 == 0 mod 13, no multiple of 6 will be in this sequence. - _Timothy L. Tiffin_, Aug 07 2016

%F a(n) = A157006(n)/2. - _Robert Price_, Oct 17 2015

%e From _Timothy L. Tiffin_, Aug 09 2016: (Start)

%e a(1) = 1, since 4^1 + 25 = 4 + 25 = 29, which is prime.

%e a(2) = 2, since 4^2 + 25 = 16 + 25 = 41, which is prime.

%e a(3) = 3, since 4^3 + 25 = 64 + 25 = 89, which is prime.

%e a(4) = 4, since 4^4 + 25 = 256 + 25 = 281, which is prime.

%e a(5) = 5, since 4^5 + 25 = 1024 + 25 = 1049, which is prime.

%e a(6) = 10, since 4^10 + 25 = 1048576 + 25 = 1048601, which is prime. (End)

%t Select[Range[2000], PrimeQ[4^# + 25] &] (* _T. D. Noe_, Feb 03 2012 *)

%o (PARI) for(n=1, 1e5, if(isprime(4^n + 25), print1(n", "))) \\ _Altug Alkan_, Oct 17 2015

%o (Magma) [n: n in [1..1000] | IsPrime(4^n+25)]; // _Vincenzo Librandi_, Aug 07 2016

%Y Cf. A057196, A104072, A157006.

%K nonn

%O 1,2

%A _Juri-Stepan Gerasimov_, Jan 15 2012

%E a(22)-a(39) derived from A157006 by _Robert Price_, Oct 17 2015

%E a(40)-a(48) derived from A157006 by _Elmo R. Oliveira_, Nov 28 2023