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.

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..10080

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

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

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

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

Cf. also A051709, A345001.

Cf. A013661, A082020.

Sequence in context: A083456 A211932 A213642 * A353860 A303073 A068058

Adjacent sequences: A344750 A344751 A344752 * A344754 A344755 A344756

Keyword

nonn,easy

Author

Antti Karttunen, May 28 2021

Extensions

New primary definition added by Antti Karttunen, Jun 06 2021

Status

approved