A129283 a(n) = (arithmetic derivative of n) + n.

0, 1, 3, 4, 8, 6, 11, 8, 20, 15, 17, 12, 28, 14, 23, 23, 48, 18, 39, 20, 44, 31, 35, 24, 68, 35, 41, 54, 60, 30, 61, 32, 112, 47, 53, 47, 96, 38, 59, 55, 108, 42, 83, 44, 92, 84, 71, 48, 160, 63, 95, 71, 108, 54, 135, 71, 148, 79, 89, 60, 152, 62, 95, 114, 256, 83, 127, 68

Offset: 0

Author

Reinhard Zumkeller, Apr 07 2007

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003415, A051674, A129284, A129285, A129286, A136141.

Programs

Haskell

a129283 n = a003415 n + n  -- Reinhard Zumkeller, Nov 01 2013

Maple

A129283 := proc(n)
    n+A003415(n) ;
end proc:
seq(A129283(n),n=0..40) ; # R. J. Mathar, Feb 04 2022

Mathematica

ad[n_] := Switch[n, 0|1, 0, ?PrimeQ, 1, , Sum[Module[{p, e}, {p, e} = pe; n*e/p], {pe, FactorInteger[n]}]];
a[n_] := ad[n] + n;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 23 2023 *)

Formula

a(n) = A003415(n) + n.

a(n) = A003415(n*A051674(k)) / A051674(k).

a(A129284(n)) > 1, a(A129285(n)) > 1, a(A129286(n)) > 1.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + Sum_{p prime} 1/(p*(p-1)) = 1 + A136141 = 1.773156... . - Amiram Eldar, May 14 2025