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%I #34 Jan 26 2020 12:42:36
%S 1,2,28,80,82,126,242,728,2400,3374,6562,6858,14640,19682,24390,28560,
%T 29790,50626,50652,59050,91126,161052,194480,194482,250048,274626,
%U 300762,328510,357912,371292,571788,707280,753570,759376,823542,970298,1157626,1295028,1442898,1771560,1860868,2146688,2146690
%N Numbers k such that either (a) k-1=i^m for some i and m >= 3 and k+1 is a prime, or (b) k-1 is a prime and k+1 = i^m for some i and m >= 3.
%C If 0 or 1 are not counted as powers, then this sequence starts with 28.
%C All terms other than 1 are even and follow or precede an odd power.
%H S. Brunner, <a href="/A329595/b329595.txt">Table of n, a(n) for n = 1..1100</a>
%e The first 20 terms with their neighbors:
%e n k-1 k k+1 | n k-1 k k+1
%e 1 0^3 1 2 | 11 3^8 6562 6563
%e 2 1^3 2 3 | 12 6857 6858 19^3
%e 3 3^3 28 29 | 13 14639 14640 11^4
%e 4 79 80 3^4 | 14 19681 19682 3^9
%e 5 3^4 82 83 | 15 29^3 24390 24391
%e 6 5^3 126 127 | 16 28559 28560 13^4
%e 7 241 242 3^5 | 17 29789 29790 31^3
%e 8 727 728 3^6 | 18 15^4 50626 50627
%e 9 2399 2400 7^4 | 19 50651 50652 37^3
%e 10 3373 3374 15^3 | 20 3^10 59050 59051
%t {1, 2}~Join~Flatten@ Map[Which[AllTrue[{#2, #3}, # > 2 &], #1 + {-1, 1}, #2 > 2, #1 - 1, #3 > 2, #1 + 1, True, Nothing] & @@ Prepend[Map[GCD @@ FactorInteger[#][[All, -1]] &, {# - 2, # + 2}], #] &, Prime@ Range[160000]] (* _Michael De Vlieger_, Dec 27 2019 *)
%o (PARI) isok(k) = (k==1) || (k==2) || ((ispower(k-1) >= 3) && isprime(k+1)) || (isprime(k-1) && (ispower(k+1) >= 3)); \\ _Michel Marcus_, Nov 18 2019
%Y Cf. A163492, A329582.
%K nonn
%O 1,2
%A _S. Brunner_, Nov 17 2019
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