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A145398
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a(n) = Sum_{d|n} sigma(d) - Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).
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3
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1, 3, 5, 11, 7, 15, 9, 31, 18, 21, 13, 55, 15, 27, 35, 75, 19, 54, 21, 77, 45, 39, 25, 155, 38, 45, 58, 99, 31, 105, 33, 167, 65, 57, 63, 198, 39, 63, 75, 217, 43, 135, 45, 143, 126, 75, 49, 375, 66, 114, 95, 165, 55, 174, 91, 279, 105, 93, 61, 385, 63, 99, 162, 355, 105, 195
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OFFSET
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1,2
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COMMENTS
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Dirichlet convolution of [1,-1,0,4,0,0,...] with A007429.
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LINKS
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FORMULA
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Dirichlet g.f.: (1-1/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
Multiplicative with a(2^e) = 3*2^(e+1)-4*e-5, and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). (End)
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MAPLE
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read("transforms") ; s1 := [1, -1, 0, 4, seq(0, n=1..40)] ; s2 := [seq(add(sigma(d), d=divisors(n)), n=1..40)] ; DIRICHLET(s1, s2) ; # R. J. Mathar, Feb 07 2011
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MATHEMATICA
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f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 3*2^(e + 1) - 4*e - 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 3*2^(f[i, 2]+1) - 4*f[i, 2] - 5, (f[i, 1]*(f[i, 1]^(f[i, 2]+1)-1) - (f[i, 1]-1)*(f[i, 2]+1))/(f[i, 1]-1)^2)); } \\ Amiram Eldar, Oct 25 2022
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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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