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A347031
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a(n) = 1 - Sum_{k=2..n} (-1)^k * a(floor(n/k)).
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4
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1, 0, 1, 1, 2, -1, 0, 0, 2, -1, 0, 2, 3, 0, 3, 3, 4, -4, -3, -1, 2, -1, 0, 0, 2, -1, 3, 5, 6, -7, -6, -6, -3, -6, -3, 7, 8, 5, 8, 8, 9, -4, -3, -1, 7, 4, 5, 5, 7, -1, 2, 4, 5, -15, -12, -12, -9, -12, -11, 7, 8, 5, 13, 13, 16, 3, 4, 6, 9, -4, -3, -7, -6, -9, -1
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,5
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COMMENTS
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LINKS
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FORMULA
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G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x - Sum_{k>=2} (-1)^k * (1 - x^k) * A(x^k)).
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MATHEMATICA
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a[n_] := a[n] = 1 - Sum[(-1)^k a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 75}]
nmax = 75; A[_] = 0; Do[A[x_] = (1/(1 - x)) (x - Sum[(-1)^k (1 - x^k) A[x^k], {k, 2, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
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PROG
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(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
if n <= 1:
return n
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (j2-j&1)*(1 if j&1 else -1)*A347031(k1)
j, k1 = j2, n//j2
return c+(n+1-j&1)*(1 if j&1 else -1) # Chai Wah Wu, Apr 04 2023
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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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