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a(n) = b(2n-1)^b(2n) where b(3n+1) = floor(n/9) + 2, b(3n+2) = (n mod 9) + 2, b(3n+3) = b(3n+1)*b(3n+2) for n >= 0.
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%I #14 Nov 01 2022 18:17:31

%S 4,16,729,16,64,9765625,64,144,678223072849,256,256,

%T 150094635296999121,1024,8000,64,27,729,16777216,243,3375,

%U 101559956668416,2187,9261,4722366482869645213696,19683,19683,1000000000000000000000000000000,16

%N a(n) = b(2n-1)^b(2n) where b(3n+1) = floor(n/9) + 2, b(3n+2) = (n mod 9) + 2, b(3n+3) = b(3n+1)*b(3n+2) for n >= 0.

%C Old name was: (2*2=4, 2*3=6, 2*4=8, 2*5=10, 2*6=12, 2*7=14, 2*8=16, 2*9=18, 2*10=20, 3*2=6, ...) becomes (abs(2^2, 4^2, 3^6, 2^4, 8^2, 5^10, 2^6, 12^2, 7^14, 2^8, 16^2, 9^18, 2^10, 20^3, 2^6, ...)).

%C (..., 9*9=81, 9*10=90, 10*2=20, 10*3=30, 10*4=40, 10*5=50, 10*6=60, 10*7=70, 10*8=80, 10*9=90, 10*10=100, 11*2=22, ...) becomes

%C (abs(..., 9^9, 81^9, 10^90, 10^2, 20^10, 3^30, 10^4, 40^10, 5^50, 10^6, 60^10, 7^70, 10^8, 80^10, 9^90, 10^10, 100^11, 2^22, ...)).

%e 2^2 = 4 = a(1),

%e 4^2 = 16 = a(2),

%e 3^6 = 729 = a(3),

%e 2^4 = 16 = a(4),

%e 8^2 = 64 = a(5),

%e 5^10 = 9765625 = a(6), etc.

%t Power @@@ Partition[Flatten@ Table[{n, k, n k}, {n, 2, 10}, {k, 2, 10}], 2, 2] (* _Michael De Vlieger_, Oct 24 2022 *)

%o (PARI) a(n) = my(k=ceil(n/27), r=n-27*(k-1), v=[]); for(i=2, 10, v=concat(v,[2*k,i,2*k*i])); for(i=2, 10, v=concat(v,[2*k+1,i,(2*k+1)*i])); v[2*r-1] ^ v[2*r] \\ _Jianing Song_, Oct 24 2022

%Y Cf. A144176, A144178.

%K nonn,less

%O 1,1

%A _Juri-Stepan Gerasimov_, Nov 19 2008

%E More terms from _R. J. Mathar_, Apr 29 2010

%E New name from _Jianing Song_, Nov 01 2022