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A182139
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Inverse Moebius transform of A061142.
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3
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1, 3, 3, 7, 3, 9, 3, 15, 7, 9, 3, 21, 3, 9, 9, 31, 3, 21, 3, 21, 9, 9, 3, 45, 7, 9, 15, 21, 3, 27, 3, 63, 9, 9, 9, 49, 3, 9, 9, 45, 3, 27, 3, 21, 21, 9, 3, 93, 7, 21, 9, 21, 3, 45, 9, 45, 9, 9, 3, 63, 3, 9, 21, 127, 9, 27, 3, 21, 9, 27, 3, 105, 3, 9, 21, 21, 9
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OFFSET
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1,2
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COMMENTS
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a(n) is multiplicative with a(p^e) = -1 + 2^(e+1).
If s is squarefree then a(s) = A048691(s).
More generally: Let a_q(n) be multiplicative with a_q(p^e) = (q^(e+1)-1)/ (q-1) for prime p, e >= 0 and some fixed integer q. Then a_q(n) is the inverse Moebius transform of the completely multiplicative sequence b_q(n) = q^bigomega(n) with b_q(p) = q and b_q(1) = 1. For q = 1 see a_q(n) = A000005(n) and b_q(n) = A000012(n), for q = 0 see a_q(n) = A000012(n) and b_q(n) = A000007(n) with offset 1, and for q = -1 see a_q(n) = A010052(n) with offset 1 and b_q(n) = A008836(n). - Werner Schulte, Feb 20 2019
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LINKS
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FORMULA
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a(n) = Sum_{d|n} 2^Omega(d) = Sum_{d|n} 2^A001222(d) = Sum_{d|n} A061142(d).
Dirichlet g.f.: zeta(s)^3 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2).
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EXAMPLE
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a(12) = a(2^2 * 3^1) = (-1 + 2^(2+1)) * (-1 + 2^(1+1)) = 7 * 3 = 21; or, using the divisors set {1,2,3,4,6,12}: 2^0 + 2^1 + 2^1 + 2^2 + 2^2 + 2^3 = 21.
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MATHEMATICA
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t[n_] := DivisorSum[n, 2^PrimeOmega[#]&]; Table[t[n], {n, 100}]
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PROG
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(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)/(1 - 2*X))[n], ", ")) \\ Vaclav Kotesovec, Mar 14 2023
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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