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A181563
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Almost-Liouville function.
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1
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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
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OFFSET
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1,8
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COMMENTS
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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.
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REFERENCES
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B. Cloitre, Almost Liouville functions, in preparation, 2011.
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LINKS
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FORMULA
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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.
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PROG
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(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;
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);
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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