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A376408 a(0) = 1, and for n > 0, a(n) = a(n-1) * A019565(a(n-1)), where A019565 is the base-2 exp-function.

%I A376408 #13 Nov 08 2024 17:58:29
%S A376408 1,2,6,90,353430,274407373885179150,
%T A376408 2443417474326613595267894539584266773823049253134356678751627846400290750
%N A376408 a(0) = 1, and for n > 0, a(n) = a(n-1) * A019565(a(n-1)), where A019565 is the base-2 exp-function.
%C A376408 a(7) has 407 digits, and a(8) has 2804 digits.
%C A376408 Like A376406, this satisfies A048675(a(n)) = a(n-1) + A048675(a(n-1)), for all n >= 1, that is, applying A048675 to the terms gives the partial sums shifted right once, A376409. However, unlike A376406, this is not a subsequence of A005117: a(3) = 90 is the first term that is not squarefree. Neither can we say that this is the lexicographically largest of such sequences, as there are also infinite sequences that begin as 1, 2, 6, 120, 38, ... or as 1, 2, 6, 120, 2042040, ... that satisfy the same condition.
%H A376408 Antti Karttunen, <a href="/A376408/b376408.txt">Table of n, a(n) for n = 0..7</a>
%H A376408 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%o A376408 (PARI)
%o A376408 A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
%o A376408 A376408(n) = if(!n,1,my(x=A376408(n-1)); x*A019565(x));
%Y A376408 Cf. A019565, A048675, A376406.
%Y A376408 Cf. A376409 (= A048675(a(n)), also partial sums from its second term onward).
%Y A376408 Cf. also analogous sequences A002110 (for A276086) and A376400 (for A276076).
%K A376408 nonn
%O A376408 0,2
%A A376408 _Antti Karttunen_, Nov 04 2024