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A344753
a(n) = sigma(n) + psi(n) - 2n = Sum_{d|n, d<n} d+(mu(n/d)^2 * d), where mu is Möbius mu-function.
11
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, 48, 2, 84, 2, 47, 30, 40, 26, 91, 2, 44, 34, 82, 2, 108, 2, 68, 60, 52, 2, 124, 15, 83, 42, 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
OFFSET
1,2
COMMENTS
Sigma is the sum of divisors (A000203), and psi is Dedekind psi-function (A001615). Coincides with the latter only on perfect numbers (A000396).
FORMULA
a(n) = Sum_{d|n, d<n} d+(A008966(n/d) * d).
a(n) = A001065(n) + A306927(n).
a(n) = A001615(n) - A033879(n).
a(n) = A344705(n) + 2*A001065(n) - n.
For squarefree n, a(n) = 2*A001065(n).
a(n) = A344997(n) / A173557(n) = A344998(n) / A342001(n). - Antti Karttunen, Jun 06 2021
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
MATHEMATICA
a[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)
PROG
(PARI) A344753(n) = sumdiv(n, d, (d<n)*(d+(issquarefree(n/d) * d)));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, May 28 2021
EXTENSIONS
New primary definition added by Antti Karttunen, Jun 06 2021
STATUS
approved