A123669 Smallest generalized Fermat prime of the form (2n)^(2^k) + 1, where k>0; or -1 if no such prime exists.

5, 17, 37, -1, 101

Offset

1,1

Comments

a(n) = -1 for n = {4, 16, 32, 64, 108, 256, 500, ...}.

All primes of the form 4*n^2 + 1 belong to a(n). They are listed in A121326(n).

Last digit of a(5k)>0 is 1.

Last digit of a(n)>0 is 7 if k is not congruent to 0 mod 5, except a(1) = 5.

All currently known a(n) for 6<n<100 are listed below:

a(7)-a(8) = {197, 257}. a(10) = 401.

a(12)-a(18) = {577, 677, 614657, 185302018885184100000000000000000000000000000001, -1, 1336337, 1297}.

a(20) = 1601. a(22)-a(24) = {197352587024076973231046657, 4477457, 5308417}.

a(27)-a(28) = {2917, 3137}. a(32)-a(33) = {-1, 4357}.

a(37)-a(38) = {5477, 1238846438084943599707227160577}. a(40)-a(42) = {40960001, 45212177, 7057}.

a(44)-a(45) = {59969537, 8101}. a(47)-a(48) = {8837, 2708192040014184559945134363758220403329915059847434832829218817}.

a(51) = 355149324327687480512960334807820417442703411649746143408158197478603636302066719166373229531510062746472251495292613758147362817.

a(53) = 126247697.

a(55)-a(60) = {12101, 375817263084708503965641077546115954135779496817219617550715846657, 662148260948741787228316709317924977225312314678010411233675575297, 13457, 193877777, 14401}.

a(62)-a(67) = {153777, 15877, -1, 16901, 303595777, 17957}.

a(70)-a(71) = {384160001, 406586897}. a(73) = 21317.

a(75)-a(82) = {22501, 284936905588473857, 562448657, 24337, 150838912030874130174020868290707457, 25601, 2564253345083631031816684000763319514758972657894465952263290175258003723567069899841752707150583949000981132009709206360818037538528413351937, 723394817}.

a(85) = 28901. a(87)-a(88) = {916636177, 30977}. a(90) = 32401.

a(92) = 33857.

a(94)-a(95) = {2435149272410363768730097404205858817, 4791383378576850493153910080681360672521575296790233332710625780023370220270083429409686634957161195934369337557766908660231890537157173340981965932463779247224064100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001}.

a(97) = 1416468497. a(99) = 1536953617.

a(n) is currently unknown for n = {6, 9, 11, 19, 21, 25, 26, 29, 30, 31, 34, 35, 36, 39, 43, 46, 49, 50, 52, 54, 61, 68, 69, 72, 74, 83, 84, 86, 89, 91, 93, 96, 98, 100, ...}.

Links

Table of n, a(n) for n=1..5.

Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.

Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Crossrefs

Cf. A121326.

Sequence in context: A146231 A186029 A083048 * A121325 A256970 A145689

Adjacent sequences: A123666 A123667 A123668 * A123670 A123671 A123672

Keyword

hard,more,sign

Author

Alexander Adamchuk, Nov 15 2006

Status

approved