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.

1, 2, 6, 14, 330, 10166, 12075690, 1174153011328084322, 73582975079922326904310062621361286633125176265747127754

Offset

0,2

Comments

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

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.

Compare also to A376408.

Links

Antti Karttunen, Table of n, a(n) for n = 0..9

Index entries for sequences related to binary expansion of n

Formula

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

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.

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

Example

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):
a(0)                               = 1          -> 0
a(1) = A019565(1)                  = 2,         -> 1     = 1
a(2) = A019565(1+2)                = 6,         -> 3     = 1+2
a(3) = A019565(1+2+6)              = 14,        -> 9     = 1+2+6
a(4) = A019565(1+2+6+14)           = 330,       -> 23    = 1+2+6+14
a(5) = A019565(1+2+6+14+330)       = 10166,     -> 353   = 1+2+6+14+330
a(6) = A019565(1+2+6+14+330+10166) = 12075690,  -> 10519 = 1+2+6+14+330+10166
etc.

Prog

(PARI)
up_to = 12;
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
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); };
v376406 = A376406list(1+up_to);
A376406(n) = v376406[1+n];

Crossrefs

Cf. A019565, A048675, A097248, A376408.

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

Subsequence of A005117.

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

Sequence in context: A131518 A222201 A130642 * A133933 A297574 A349984

Adjacent sequences: A376403 A376404 A376405 * A376407 A376408 A376409

Keyword

nonn

Author

Antti Karttunen, Nov 04 2024

Status

approved