A068719 Arithmetic derivative of even numbers: a(n) = n+2*A003415(n).

1, 4, 5, 12, 7, 16, 9, 32, 21, 24, 13, 44, 15, 32, 31, 80, 19, 60, 21, 68, 41, 48, 25, 112, 45, 56, 81, 92, 31, 92, 33, 192, 61, 72, 59, 156, 39, 80, 71, 176, 43, 124, 45, 140, 123, 96, 49, 272, 77, 140, 91, 164, 55, 216, 87, 240, 101, 120

Offset: 1

Author

Reinhard Zumkeller, Feb 26 2002

Keywords

• nonn
• easy

Comments

Terms are either odd or multiples of 4. - Antti Karttunen, Jul 31 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Bruno Berselli)

Crossrefs

Cf. A003415, A005843, A068720, A068721, A260624, A327862, A327864.

Second diagonal (without the initial 1) in A084890.

Row 1 of A344027.

Programs

Magma

Ad:=func; [Ad(2*n): n in [1..60]]; // Bruno Berselli, Oct 22 2013

Mathematica

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; a[n_] := ad[2*n]; Array[a, 100] (* Amiram Eldar, Apr 11 2025 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A068719(n) = (n + 2*A003415(n)); \\ Antti Karttunen, Jul 31 2022

Formula

a(n) = A003415(A005843(n)).