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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.

%I A344705 #20 Dec 08 2023 12:44:26
%S A344705 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,
%T A344705 23,20,29,30,31,17,33,34,35,17,37,38,39,22,41,42,43,32,39,46,47,20,48,
%U A344705 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
%N 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.
%C A344705 First negative term occurs as a(1440) = -18.
%H A344705 Antti Karttunen, <a href="/A344705/b344705.txt">Table of n, a(n) for n = 1..16560</a>
%H A344705 Antti Karttunen, <a href="/A344705/a344705.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A344705 a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).
%F A344705 a(n) = A033879(n) + A306927(n).
%F A344705 a(n) = n + A344753(n) - 2*A001065(n).
%F A344705 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
%t A344705 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 *)
%o A344705 (PARI)
%o A344705 A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
%o A344705 A344705(n) = ((n + A001615(n)) - sigma(n));
%Y A344705 Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.
%Y A344705 Cf. A013661, A082020.
%K A344705 sign,easy
%O A344705 1,2
%A A344705 _Antti Karttunen_, May 28 2021