A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

-1, 0, -1, 3, -3, 5, -5, 11, 1, 5, -9, 20, -11, 5, 2, 31, -15, 24, -17, 26, 0, 5, -21, 56, -9, 5, 13, 32, -27, 43, -29, 79, -4, 5, -10, 79, -35, 5, -6, 78, -39, 53, -41, 44, 27, 5, -45, 140, -27, 38, -10, 50, -51, 93, -22, 100, -12, 5, -57, 140, -59, 5, 29, 191, -28, 73, -65, 62, -16, 63, -69, 207, -71, 5, 29, 68

Offset

1,4

Comments

Coincides with A003415 only on perfect numbers (A000396).

Links

Table of n, a(n) for n=1..76.

Formula

a(n) = A003415(n) - A033879(n) = A000203(n) + A003415(n) - 2*n.

a(n) = A001065(n) + A168036(n).

a(n) = A344999(n) / A048250(n) = A345049(n) / A173557(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 + A136141 - 2 = 0.418090735898... . - Amiram Eldar, Dec 08 2023

Mathematica

A003415[n_] := If[n < 2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
a[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
Array[a, 80] (* Jean-François Alcover, Jun 12 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));

Crossrefs

Cf. A000203, A000396, A001065, A003415, A033879, A168036, A344999, A345002, A345003, A345004, A345005, A345049.

Cf. also A051709, A344753 (analogous sequences).

Cf. A013661, A136141.

Sequence in context: A283843 A241508 A099536 * A082434 A078752 A343045

Adjacent sequences: A344998 A344999 A345000 * A345002 A345003 A345004

Keyword

sign,easy

Author

Antti Karttunen, Jun 05 2021

Status

approved