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A191414
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Unitary Jordan function J_2^*(n).
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3
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1, 3, 8, 15, 24, 24, 48, 63, 80, 72, 120, 120, 168, 144, 192, 255, 288, 240, 360, 360, 384, 360, 528, 504, 624, 504, 728, 720, 840, 576, 960, 1023, 960, 864, 1152, 1200, 1368, 1080, 1344, 1512, 1680, 1152, 1848, 1800, 1920, 1584, 2208, 2040, 2400, 1872, 2304, 2520, 2808, 2184, 2880, 3024, 2880, 2520
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OFFSET
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1,2
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COMMENTS
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Unitary Jordan functions J_k^*(n) are the unitary convolution of the unitary Mobius function and n^k, or simply J_k^*(n) = J_1^*(n^k) with J_1^*(n) = A047994(n). They are multiplicative with a(p^e) = p^(k*e)-1.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(2*e)-1, e>0.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(d) * (n/d)^2. - Amiram Eldar, Sep 12 2020
Sum_{k>=1} 1/a(k) = 1.7789153256588699707937240866939851480088485084691145802685706798681731662... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.298388788003... . - Amiram Eldar, Nov 05 2022
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MATHEMATICA
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a[n_] := Times @@ (#[[1]]^#[[2]] - 1 & ) /@ FactorInteger[n^2]; a[1] = 1; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); } \\ Amiram Eldar, Nov 05 2022
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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