A376406 - OEIS

%I #34 Nov 08 2024 17:58:18

%S 1,2,6,14,330,10166,12075690,1174153011328084322,

%T 73582975079922326904310062621361286633125176265747127754

%N 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 a(9) has 272 digits and a(10) has 1523 digits.

%C 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 Compare also to A376408.

%H Antti Karttunen, <a href="/A376406/b376406.txt">Table of n, a(n) for n = 0..9</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(n) = A019565(A376407(n)) = A019565(Sum_{i=0..n-1} a(i)).

%F 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 For n > 0, a(n) <= a(n-1) * A019565(a(n-1)).

%e 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 a(0)                               = 1          -> 0

%e a(1) = A019565(1)                  = 2,         -> 1     = 1

%e a(2) = A019565(1+2)                = 6,         -> 3     = 1+2

%e a(3) = A019565(1+2+6)              = 14,        -> 9     = 1+2+6

%e a(4) = A019565(1+2+6+14)           = 330,       -> 23    = 1+2+6+14

%e a(5) = A019565(1+2+6+14+330)       = 10166,     -> 353   = 1+2+6+14+330

%e a(6) = A019565(1+2+6+14+330+10166) = 12075690,  -> 10519 = 1+2+6+14+330+10166

%e etc.

%o (PARI)

%o up_to = 12;

%o A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };

%o 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 v376406 = A376406list(1+up_to);

%o A376406(n) = v376406[1+n];

%Y Cf. A019565, A048675, A097248, A376408.

%Y Cf. A376407 (= A048675(a(n)), also gives the partial sums from its second term onward).

%Y Subsequence of A005117.

%Y Cf. also analogous sequences A002110 (for A276085), A093502 (for A056239), A376399 (for A276075).

%K nonn

%O 0,2

%A _Antti Karttunen_, Nov 04 2024