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a(1) = 1, a(2) = 2. For n > 2, a(n) is the least novel power of the greatest prime divisor of a(n-2) + a(n-1).
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%I #17 Jan 29 2023 10:55:34

%S 1,2,3,5,4,9,13,11,27,19,23,7,25,8,121,43,41,49,125,29,1331,17,337,59,

%T 14641,343,1873,277,1849,1063,169,161051,2687,81869,21139,37,2647,61,

%U 677,1681,131,151,47,1771561,761,295387,1369,74189,257,37223,937,53,19487171

%N a(1) = 1, a(2) = 2. For n > 2, a(n) is the least novel power of the greatest prime divisor of a(n-2) + a(n-1).

%C In other words, a(n) is the least novel number p^k, where p is the greatest prime divisor of a(n-2) + a(n-1) and k >= 1. All terms are prime powers, though it is not known if every such number appears. Primes to the first power do not occur in natural order (e.g., 43 precedes 29).

%H Michael De Vlieger, <a href="/A359874/a359874.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing and labeling primes in red.

%e a(3) = 3 because a(1) + a(2) = 3, and no power of 3 has yet appeared.

%e a(4) = 5 because a(2) + a(3) = 5, and no power of 5 has yet appeared.

%e a(15) = 121 because a(13) + a(14) = 33 which has greatest prime divisor 11. But 11^1 has already occurred at a(8), so the least novel power of 11 is 11^2 = 121.

%t nn = 120; c[_] = False; q[_] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = FactorInteger[a[1] + a[2]][[-1, 1]]; u = 2; Do[k = q[s]; While[c[s^k], k++]; While[c[s^q[s]], q[s]++]; k = s^k; Set[{a[n], c[k], s}, {k, True, FactorInteger[a[n - 1] + k][[-1, 1]]}], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, Jan 16 2023 *)

%Y Cf. A000961.

%K nonn

%O 1,2

%A _David James Sycamore_, Jan 16 2023

%E More terms from _Michael De Vlieger_, Jan 16 2023