A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).

0, -1, -1, -2, 0, -4, -1, -6, 4, -3, -3, -10, 4, -12, -5, -7, 16, -16, 3, -18, 4, -11, -9, -22, 20, -15, -11, 0, 4, -28, 1, -30, 48, -19, -15, -23, 24, -36, -17, -23, 28, -40, -1, -42, 4, -6, -21, -46, 64, -35, -5, -31, 4, -52, 27, -39, 36, -35, -27, -58, 32, -60, -29

Offset: 0

Author

Paolo P. Lava, Nov 17 2009

Keywords

• easy
• sign

Comments

Let k = n'-n. For k = -1 n is a primary pseudoperfect number (A054377), apart from n=1; For k=0 n is p^p, being p a prime number (A051674); For k = 1 n is a Giuga number (A007850).

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A007850, A051674, A054377, A136141.

Cf. A003415, A051674, A083347, A083348.

Programs

Haskell

a168036 n = a003415 n - n  -- Reinhard Zumkeller, May 22 2015

Maple

with(numtheory);
A168036:=proc(q)
local n,p;
for n from 0 to q do
  print(n*add(op(2,p)/op(1,p),p=ifactors(n)[2])-n); od; end:
A168036(1000); # Paolo P. Lava, Nov 05 2012

Mathematica

np[k_] := Module[{f, n, m, p}, If[k < 2, np[k] = 0; Return[0], If[PrimeQ[k], np[k] = 1; Return[1], f = FactorInteger[k, 2]; m = f[[1, 1]]; n = k/m; p = m np[n] + n np[m]; np[k] = p; Return[p]]]];
Table[np[n] - n, {n, 0, 100}] (* Robert Price, Mar 14 2020 *)

Formula

a(A083347(n)) < 0; a(A051674(n)) = 0; a(A083348(n)) > 0. - Reinhard Zumkeller, May 22 2015

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = -1 + Sum_{p prime} 1/(p*(p-1)) = A136141 - 1 = -0.226843... . - Amiram Eldar, Dec 08 2023