A296074 - OEIS

%I #11 Dec 04 2023 01:39:26

%S 0,1,1,2,1,4,1,3,3,6,1,5,1,8,7,4,1,9,1,9,9,12,1,2,5,14,8,13,1,16,1,5,

%T 13,18,11,3,1,20,15,8,1,24,1,21,18,24,1,-9,7,27,19,25,1,20,15,14,21,

%U 30,1,-1,1,32,24,6,17,40,1,33,25,40,1,-27,1,38,32,37,17,48,1,-1,22,42,1,9,21,44,31,26,1,18,19,45,33,48,23,-36,1,53,36,33

%N Sum of deficiencies of the proper divisors of n.

%H Antti Karttunen, <a href="/A296074/b296074.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = Sum_{d|n, d<n} A033879(d).

%F a(n) = A296075(n) - A033879(n).

%F Sum_{k=1..n} a(k) ~ (Pi^2/4 - Pi^4/72 - 1) * n^2. - _Amiram Eldar_, Dec 04 2023

%e For n = 6, whose proper divisors are 1, 2, 3, their deficiencies are 1, 1, 2, thus a(6) = 1+1+2 = 4.

%e For n = 12, whose proper divisors are 1, 2, 3, 4, 6, their deficiencies are 1, 1, 2, 1, 0, thus a(12) = 1+1+2+1+0 = 5.

%t f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2; a[1] = 0; a[n_] := Module[{f = FactorInteger[n]}, 3 * Times @@ f1 @@@ f - Times @@ f2 @@@ f - 2*n]; Array[a, 100] (* _Amiram Eldar_, Dec 04 2023 *)

%o (PARI)

%o A033879(n) = ((2*n)-sigma(n));

%o A296074(n) = sumdiv(n,d,(d<n)*A033879(d));

%Y Cf. A033879.

%Y Cf. also A294886, A294887, A294888, A294889, A293438 (product of).

%K sign,easy

%O 1,4

%A _Antti Karttunen_, Dec 04 2017