A348975 a(n) = A003415(n) + A003958(n) - n, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

0, 0, 0, 1, 0, 1, 0, 5, 1, 1, 0, 6, 0, 1, 1, 17, 0, 7, 0, 8, 1, 1, 0, 22, 1, 1, 8, 10, 0, 9, 0, 49, 1, 1, 1, 28, 0, 1, 1, 32, 0, 11, 0, 14, 10, 1, 0, 66, 1, 11, 1, 16, 0, 35, 1, 42, 1, 1, 0, 40, 0, 1, 12, 129, 1, 15, 0, 20, 1, 13, 0, 88, 0, 1, 12, 22, 1, 17, 0, 100, 43, 1, 0, 52, 1, 1, 1, 62, 0, 49, 1, 26, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn
• look

Comments

No negative terms. See comments in A322582.

This is the difference between the arithmetic derivative of n [= A003415(n)] and its guaranteed lower bound A322582(n) [= n - A003958(n)].

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A003415, A003958, A168036, A322582.

Cf. also A348970 for the corresponding difference from a guaranteed upper bound.

Programs

Mathematica

MapAt[# + 1 &, Array[If[# < 2, 0, # Total[#2/#1 & @@@ #2]] + Times @@ Map[(#1 - 1)^#2 & @@ # &, #2] - #1 & @@ {#, FactorInteger[#]} &, 95], 1] (* Michael De Vlieger, Mar 15 2022 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348975(n) = (A003415(n) - A322582(n));

Formula

a(n) = A003415(n) - A322582(n).

a(n) = A003958(n) + A168036(n).