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Smallest prime of the form (k^2 * 2^n + 1).

1

`%I #4 Aug 31 2020 09:03:08
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`%S 3,5,73,17,1153,257,1153,257,18433,25601,18433,65537,1179649,65537,
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`%T 1179649,65537,1179649,26214401,117964801,26214401,169869313,
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`%U 104857601,2717908993,10485760001,2717908993,11341398017,10871635969,52613349377
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`%N Smallest prime of the form (k^2 * 2^n + 1).
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`%C It is interesting to note a pattern such that for many n a(n) = a(n+2) and a(n+1) = a(n+3). The first such double twin pair run starts at n = 5, a(5) = a(7) = 1153 and a(6) = a(8) = 257. The first triple twin pair run starts at n = 12, a(12) = a(14) = a(16) = 65537 and a(13) = a(15) = a(17) = 1179649. There are longer runs of twin pairs such as penta twin pair run starting at n = 55, a(55) = a(57) = a(59) = a(61) = a(63) = 83010348331692982273 and a(56) = a(58) = a(60) = a(62) = a(64) = 461168601842738790401. A run of six twins starts at n = 71, a(71) = a(73) = a(75) = a(77) = a(79) = a(81) = 21760664753063325144711169. The final index of many twin runs is a perfect power such as {8,16,64,81,...}. Corresponding minimum numbers k such that (k^2*2^n + 1) is prime are listed in A122913[n] = { 1,1,3,1,6,2,3,1,6,5, 3,4,12,2,6,1,3,10,15,5, 9,5,18,25,9,13,9,14,12,7, 6,9,3,17,9,9,15,12,9,6, 6,3,3,11,42,18,21,9,66,10, 33,5,27,7,48,80,24,40,12,20, 6,10,3,5,3,7,3,79,75,63, 96,40,48,20,24,10,12,5,6,15, 3,22,72,11,36,15,18,25,9,57, 21,44,33,22,93,11,366,38,183,19,...}.
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`%Y Cf. A122913, A092506.
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`%K nonn
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`%O 1,1
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`%A _Alexander Adamchuk_, Sep 18 2006
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