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

1,2

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

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

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 *)

Prog

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

Crossrefs

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

Cf. A013661, A082020.

Sequence in context: A085310 A055653 A155918 * A331170 A325183 A343247

Adjacent sequences: A344702 A344703 A344704 * A344706 A344707 A344708

Keyword

sign,easy

Author

Antti Karttunen, May 28 2021

Status

approved