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A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 exp-function.

%I A376406 #34 Nov 08 2024 17:58:18
%S A376406 1,2,6,14,330,10166,12075690,1174153011328084322,
%T A376406 73582975079922326904310062621361286633125176265747127754
%N A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 exp-function.
%C A376406 a(9) has 272 digits and a(10) has 1523 digits.
%C A376406 The lexicographically earliest infinite sequence x for which A048675(x(n)) gives the partial sums of x (shifted right once). This follows because the "least k" condition in the alternative formula also ensures that each k is squarefree, as we have A097248(n) = A019565(A048675(n)) <= n for all n, with equivalence only when n is squarefree.
%C A376406 Compare also to A376408.
%H A376406 Antti Karttunen, <a href="/A376406/b376406.txt">Table of n, a(n) for n = 0..9</a>
%H A376406 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A376406 a(n) = A019565(A376407(n)) = A019565(Sum_{i=0..n-1} a(i)).
%F A376406 a(0) = 1, and for n > 0, a(n) is the least k such that A048675(k) = a(n-1) + A048675(a(n-1)), where A048675 is the base-2 log-function.
%F A376406 For n > 0, a(n) <= a(n-1) * A019565(a(n-1)).
%e A376406 Starting with a(0) = 1, we take partial sums of previous terms, and apply A019565 to get the next term, and in the rightmost column, we "unbox" that term by applying A048675 to get A376407(n), which thus gives the partial sums of a(0)..a(n-1):
%e A376406 a(0)                               = 1          -> 0
%e A376406 a(1) = A019565(1)                  = 2,         -> 1     = 1
%e A376406 a(2) = A019565(1+2)                = 6,         -> 3     = 1+2
%e A376406 a(3) = A019565(1+2+6)              = 14,        -> 9     = 1+2+6
%e A376406 a(4) = A019565(1+2+6+14)           = 330,       -> 23    = 1+2+6+14
%e A376406 a(5) = A019565(1+2+6+14+330)       = 10166,     -> 353   = 1+2+6+14+330
%e A376406 a(6) = A019565(1+2+6+14+330+10166) = 12075690,  -> 10519 = 1+2+6+14+330+10166
%e A376406 etc.
%o A376406 (PARI)
%o A376406 up_to = 12;
%o A376406 A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
%o A376406 A376406list(up_to) = { my(v=vector(up_to), s=1); v[1]=1; for(n=2,up_to,v[n] = A019565(s); s += v[n]); (v); };
%o A376406 v376406 = A376406list(1+up_to);
%o A376406 A376406(n) = v376406[1+n];
%Y A376406 Cf. A019565, A048675, A097248, A376408.
%Y A376406 Cf. A376407 (= A048675(a(n)), also gives the partial sums from its second term onward).
%Y A376406 Subsequence of A005117.
%Y A376406 Cf. also analogous sequences A002110 (for A276085), A093502 (for A056239), A376399 (for A276075).
%K A376406 nonn
%O A376406 0,2
%A A376406 _Antti Karttunen_, Nov 04 2024