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A133521 Smallest k such that p(n)^5 - k is prime where p(n) is the n-th prime.

%I A133521 #7 Sep 27 2020 11:41:00
%S A133521 1,2,4,20,4,2,18,18,16,6,2,6,24,12,36,22,10,8,8,24,20,86,22,6,18,42,
%T A133521 26,6,50,52,20,12,48,2,196,68,18,14,16,16,18,2,10,6,16,38,2,36,6,2,16,
%U A133521 42,18,42,40,34,22,2,38,4,36,52,26,132,36,28,24,74,46,36,4,16,8,24,80,16
%N A133521 Smallest k such that p(n)^5 - k is prime where p(n) is the n-th prime.
%H A133521 Harvey P. Dale, <a href="/A133521/b133521.txt">Table of n, a(n) for n = 1..1000</a>
%e A133521 p(10)=29, 29^5 = 20511149; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k.
%e A133521 for k = 2: 20511149 - 2 = 20511147, which is 3 * 23 * 297263 and not prime.
%e A133521 for k = 4: 20511149 - 4 = 20511145, which is 5 * 4102229, also not prime.
%e A133521 for k = 6: 20511149 - 6 = 20511141, which is prime, so 6 is the smallest number that can be subtracted from 20511149 to make another prime.
%e A133521 Hence a(10) = 6.
%t A133521 #-NextPrime[#,-1]&/@(Prime[Range[80]]^5) (* _Harvey P. Dale_, Sep 27 2020 *)
%Y A133521 Cf. A050997, A054271, A091666, A133517, A133518, A133519, A133520, A133522, (A001223).
%K A133521 easy,nonn
%O A133521 1,2
%A A133521 _Carl R. White_, Sep 14 2007