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Least k such that the smallest prime >= 2^k is 2^k + 2*n + 1, or -1 if no such k exists.
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%I #22 May 31 2024 14:38:14

%S 0,3,5,10,9,29,64,22,13,162,19,39,34,14,17,36,60,25,74,87,121,24,151,

%T 209,170,111,35,50,188,45,96,247,193,124,49,115,258,83,173,254,56,167,

%U 136,138,279,148,314,153,158,106,199,434,93,161,6954,104,719,240,164

%N Least k such that the smallest prime >= 2^k is 2^k + 2*n + 1, or -1 if no such k exists.

%C a(n) = -1 if 2*n + 1 is a Sierpiński number (for example when 2*n + 1 = 78557); cf. A076336. See also A067760.

%C Conjecture: a(n) != -1 if 2*n + 1 is not a Sierpiński number. In other words, if 2*n + 1 is not a Sierpiński number, then there exists some k >= 1 such that 2^k + 1, 2^k + 3, ..., 2^k + 2*n - 1 are all composite while 2^k + 2*n + 1 is prime.

%C a(54), a(75), a(83), a(128), a(159), a(176), ... > 5000 (if not equal to -1), which means that 109, 151, 167, 257, 319, 353, ... do not present among the first 5000 terms of A092131.

%C a(75) = 5880, a(83) = 5513. - _Michael S. Branicky_, May 28 2024

%C a(128) > 7000. - _Michael S. Branicky_, May 30 2024

%H Michael S. Branicky, <a href="/A373210/b373210.txt">Table of n, a(n) for n = 0..127</a>

%e a(6) = 64, because the smallest prime >= 2^k is not 2^k + 13 for 0 <= k <= 63, while the smallest prime >= 2^64 is 2^64 + 13.

%o (PARI) A373210_first_N_terms(N) = my(v = vector(N+1, i, -1), dist); v[1] = 0; for(i=2, oo, dist = nextprime(2^i) - 2^i; if(dist <= 2*N+1 && v[(dist+1)/2] == -1, v[(dist+1)/2] = i); if(vecmin(v) > -1, return(v))) \\ Warning: ignoring Sierpinski numbers

%Y Cf. A067760, A076336, A092131.

%K sign

%O 0,2

%A _Jianing Song_, May 28 2024

%E a(54) and beyond from _Michael S. Branicky_, May 29 2024