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a(n) is obtained by replacing 2's by n's in the hereditary base-2 expansion of n.
1

%I #8 Jun 07 2021 14:54:52

%S 0,1,2,4,256,3126,46662,823551,134217728,3486784402,100000000010,

%T 3138428376733,115909305827328,4240251492291543,166680102383370254,

%U 7006302246093750016

%N a(n) is obtained by replacing 2's by n's in the hereditary base-2 expansion of n.

%C The next term, a(16), has 22212093154093428530 digits, and is too large to include.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Goodstein&#39;s_theorem#Hereditary_base-n_notation">Hereditary base-n notation</a>

%F a(n) = A342707(n, n).

%e For n = 5:

%e - 5 = 2^2^2^0 + 2^0,

%e - so a(5) = 5^5^5^0 + 5^0 = 3126.

%Y Cf. A054382, A342707.

%K nonn

%O 0,3

%A _Rémy Sigrist_, Jun 04 2021