A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

1, 2, 3, 3, 5, 6, 7, 5, 8, 10, 11, 8, 13, 14, 15, 9, 17, 15, 19, 14, 21, 22, 23, 12, 24, 26, 23, 20, 29, 30, 31, 17, 33, 34, 35, 17, 37, 38, 39, 22, 41, 42, 43, 32, 39, 46, 47, 20, 48, 47, 51, 38, 53, 42, 55, 32, 57, 58, 59, 36, 61, 62, 55, 33, 65, 66, 67, 50, 69, 70, 71, 21, 73, 74, 71, 56, 77, 78, 79, 38, 68, 82

Offset: 1

Author

Antti Karttunen, May 28 2021

Keywords

• sign
• easy

Comments

First negative term occurs as a(1440) = -18.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16560
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.

Cf. A013661, A082020.

Programs

Mathematica

f[p_, e_] := (p^(e+1) - 1)/(p-1); a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, n * (Times @@ (1 + 1/fct[[;; , 1]]) + 1) - Times @@ f @@@ fct]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

PARI

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
A344705(n) = ((n + A001615(n)) - sigma(n));

Formula

a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).

a(n) = A033879(n) + A306927(n).

a(n) = n + A344753(n) - 2*A001065(n).

Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 + 1 - Pi^2/6 = 0.874883... . - Amiram Eldar, Dec 08 2023