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Add primes until a perfect power appears. When a perfect power appears, that term is the smallest root of the perfect power. Then return to adding primes, beginning with the next prime.
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%I #9 Jun 23 2023 17:04:35

%S 2,5,10,17,28,41,58,77,10,39,70,107,148,191,238,291,350,411,478,549,

%T 622,701,28,117,214,315,418,525,634,747,874,1005,1142,1281,1430,1581,

%U 1738,1901,2068,2241,2420,51,242,435,632,831,1042,1265,1492,1721,1954,2193

%N Add primes until a perfect power appears. When a perfect power appears, that term is the smallest root of the perfect power. Then return to adding primes, beginning with the next prime.

%e The first term is the first prime, p(1) = 2

%e a(1) = p(1) = 2

%e a(2) = a(1) + p(2) = 2 + 3 = 5

%e a(3) = a(2) + p(3) = 5 + 5 = 10

%e etc.

%e a(8) = 58 + 19 = 77

%e a(9) is determined:

%e a(8) + p(9) = 77 + 23 = 100, a perfect power. 10 is the smallest root of 100, therefore a(9) = 10

%e a(10) = 10 + p(10) = 10 + 29 = 39

%e and so on.

%t root[n_] := Surd[n, GCD @@ FactorInteger[n][[;; , 2]]]; a[1] = 2; a[n_] := a[n] = root[a[n - 1] + Prime[n]]; Array[a, 100] (* _Amiram Eldar_, May 21 2023 *)

%Y Cf. A001597.

%K nonn,easy

%O 1,1

%A _Damon Lay_, May 16 2023