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Möbius transform of sigma_n(n).
1

%I #38 Jun 08 2023 00:20:09

%S 1,4,27,268,3125,47418,823543,16842736,387440145,10009763520,

%T 285311670611,8918294495628,302875106592253,11112685047823702,

%U 437893920912783255,18447025552964452096,827240261886336764177,39346558271491791438000,1978419655660313589123979

%N Möbius transform of sigma_n(n).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%F a(n) = Sum_{d|n} sigma_d(d) * mu(n/d).

%e a(6) = 47418; a(6) = Sum_{d|6} sigma_d(d) * mu(6/d) = sigma_1(1) * mu(6/1) + sigma_2(2) * mu(6/2) + sigma_3(3) * mu(6/3) + sigma_6(6) * mu(6/6) = 1*1 + 5*(-1) + 28*(-1) + 47450*1 = 47418.

%t a[n_] := DivisorSum[n, DivisorSigma[#, #]*MoebiusMu[n/#] &]; Array[a, 20] (* _Wesley Ivan Hurt_, Nov 12 2022 *)

%o (PARI) a(n) = sumdiv(n, d, sigma(d,d)*moebius(n/d)); \\ _Michel Marcus_, Jun 24 2022

%Y Cf. A008683 (mu), A023887.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Jun 24 2022