%I #30 Dec 08 2023 12:31:35

%S 0,2,2,5,2,12,2,11,7,16,2,28,2,20,18,23,2,39,2,38,22,28,2,60,11,32,22,

%T 48,2,84,2,47,30,40,26,91,2,44,34,82,2,108,2,68,60,52,2,124,15,83,42,

%U 78,2,120,34,104,46,64,2,192,2,68,74,95,38,156,2,98,54,148,2,195,2,80,94,108,38,180,2,170,67,88,2

%N a(n) = sigma(n) + psi(n) - 2n = Sum_{d|n, d<n} d+(mu(n/d)^2 * d), where mu is Möbius mu-function.

%C Sigma is the sum of divisors (A000203), and psi is Dedekind psi-function (A001615). Coincides with the latter only on perfect numbers (A000396).

%H Antti Karttunen, <a href="/A344753/b344753.txt">Table of n, a(n) for n = 1..10080</a>

%H Antti Karttunen, <a href="/A344753/a344753.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = Sum_{d|n, d<n} d+(A008966(n/d) * d).

%F a(n) = A001065(n) + A306927(n).

%F a(n) = A001615(n) - A033879(n).

%F a(n) = A344705(n) + 2*A001065(n) - n.

%F For squarefree n, a(n) = 2*A001065(n).

%F a(n) = A344997(n) / A173557(n) = A344998(n) / A342001(n). - _Antti Karttunen_, Jun 06 2021

%F Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = Pi^2/6 + 15/Pi^2 - 2 = 1.164751... . - _Amiram Eldar_, Dec 08 2023

%t a[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];

%t Array[a, 100] (* _Jean-François Alcover_, Jun 12 2021 *)

%o (PARI) A344753(n) = sumdiv(n,d,(d<n)*(d+(issquarefree(n/d) * d)));

%Y Cf. A001065, A001615, A008683, A008966, A033879, A173557, A306927, A344705, A342001, A344705, A344754, A344755, A344997, A344998.

%Y Cf. A000396 [where a(n) = A001615(n)], A005100 [a(n) < A001615(n)], A005101 [a(n) > A001615(n)].

%Y Cf. also A051709, A345001.

%Y Cf. A013661, A082020.

%K nonn,easy

%O 1,2

%A _Antti Karttunen_, May 28 2021

%E New primary definition added by _Antti Karttunen_, Jun 06 2021