A296074 Sum of deficiencies of the proper divisors of n.

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, 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, 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

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

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

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

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

Example

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

Mathematica

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 *)

Prog

(PARI)
A033879(n) = ((2*n)-sigma(n));
A296074(n) = sumdiv(n, d, (d<n)*A033879(d));

Crossrefs

Cf. A033879.

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

Sequence in context: A243504 A292257 A317837 * A239451 A216223 A078072

Adjacent sequences: A296071 A296072 A296073 * A296075 A296076 A296077

Keyword

sign,easy

Author

Antti Karttunen, Dec 04 2017

Status

approved