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A341444
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Dirichlet inverse of A083399, where A083399(n) = 1 + omega(n).
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2
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1, -2, -2, 2, -2, 5, -2, -2, 2, 5, -2, -7, -2, 5, 5, 2, -2, -7, -2, -7, 5, 5, -2, 9, 2, 5, -2, -7, -2, -16, -2, -2, 5, 5, 5, 14, -2, 5, 5, 9, -2, -16, -2, -7, -7, 5, -2, -11, 2, -7, 5, -7, -2, 9, 5, 9, 5, 5, -2, 30, -2, 5, -7, 2, 5, -16, -2, -7, 5, -16, -2, -23, -2, 5, -7, -7, 5, -16, -2, -11, 2, 5, -2, 30, 5, 5, 5, 9, -2, 30, 5
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OFFSET
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1,2
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COMMENTS
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The Dirichlet inverse function, a(n) = (omega + 1)^(-1)(n). - Original name.
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LINKS
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FORMULA
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = -DivisorSum[n, (PrimeNu[n/#] + 1)*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)
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PROG
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(PARI) cOmega(n) = if (n==1, 1, my(f=factor(n)); bigomega(n)!*prod(k=1, #f~, 1/f[k, 2]!)); \\ A008480
a(n) = (-1)^bigomega(n)*sumdiv(n, d, moebius(n/d)^2*cOmega(d));
(PARI)
memoA341444 = Map();
A341444(n) = if(1==n, 1, my(v); if(mapisdefined(memoA341444, n, &v), v, v = -sumdiv(n, d, if(d<n, (1+omega(n/d))*A341444(d), 0)); mapput(memoA341444, n, v); (v))); \\ Antti Karttunen, Jul 21 2022~
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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Data section extended up to a(91) and name edited by Antti Karttunen, Jul 21 2022
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STATUS
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approved
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