A211991 Difference between the arithmetic derivative of n and the sum of proper divisors of n.

0, 0, 0, 1, 0, -1, 0, 5, 2, -1, 0, 0, 0, -1, -1, 17, 0, 0, 0, 2, -1, -1, 0, 8, 4, -1, 14, 4, 0, -11, 0, 49, -1, -1, -1, 5, 0, -1, -1, 18, 0, -13, 0, 8, 6, -1, 0, 36, 6, 2, -1, 10, 0, 15, -1, 28, -1, -1, 0, -16, 0, -1, 10, 129, -1, -17, 0, 14, -1, -15, 0, 33

Offset

1,8

Comments

Observations: at least the first 50 indices of nonnegative terms are also the first 50 terms of A212165. Also at least the first 28 indices of negative terms are also the first 28 terms of A212168, since A212168 is the complement of A212165.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = A003415(n) - A001065(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (A136141 - A013661 + 1) / 2 = 0.0641113... . - Amiram Eldar, Mar 17 2024

Mathematica

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Table[dn[n] - (DivisorSigma[1, n] - n), {n, 100}] (* T. D. Noe, Dec 27 2012 *)

Prog

(PARI)
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A211991(n) = (A003415(n) - (sigma(n)-n)); \\ Antti Karttunen, Mar 08 2018

Crossrefs

Cf. A000203, A001065, A003415.

Cf. A013661, A136141.

Sequence in context: A111560 A371026 A324609 * A354596 A303127 A106266

Adjacent sequences: A211988 A211989 A211990 * A211992 A211993 A211994

Keyword

sign

Author

Omar E. Pol, Dec 18 2012

Status

approved