A181563 Almost-Liouville function.

1, -1, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0

Offset

1,8

Comments

The partial sum a(1)+a(2)+...+a(n) is asymptotic to -sqrt(2n).

The Liouville lambda function satisfies: Sum_{k=1..n} lambda(k)*floor(n/k) = floor(n^(1/2)), that's why this sequence is "almost" a Liouville function.

References

B. Cloitre, Almost Liouville functions, in preparation, 2011.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

Sum_{k=1..n} a(k)*g(n/k) = floor(n^(1/2)) where g(x) = 2^floor(log(x)/log(2)) for n>1 with a(1)=1.

a(1) = 1; for n > 1, a(n) = A000196(n) - Sum{k=1 .. n-1} a(k)*A053644(floor(n/k)). - Antti Karttunen, Sep 28 2018

Prog

(PARI) {a(n)=if(n==1, 1, floor(n^(1/2))-sum(k=1, n-1, a(k)*2^floor(log(n/k)/log(2))))}
(PARI)
up_to = 2048;
A000196(n) = sqrtint(n);
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644. Note that we are calling this here also with rational arguments, but it works all fine.
A181563list(up_to) = { my(v=vector(up_to)); v[1] = 1; for(n=2, up_to, v[n] = A000196(n) - sum(k=1, n-1, v[k]*A053644(n/k))); (v); };
v181563 = A181563list(up_to);
A181563(n) = v181563[n]; \\ Antti Karttunen, Sep 28 2018

Crossrefs

Cf. A000196, A008836, A053644.

Sequence in context: A304021 A056974 A368844 * A056976 A279281 A342465

Adjacent sequences: A181560 A181561 A181562 * A181564 A181565 A181566

Keyword

sign

Author

Benoit Cloitre, Jan 29 2011

Status

approved