A344584 Difference between the inverse Möbius transform of the arithmetic derivative of n and the sum of the proper divisors of n: a(n) = A319684(n) - A001065(n).

0, 0, 0, 2, 0, 1, 0, 10, 3, 1, 0, 11, 0, 1, 1, 34, 0, 13, 0, 15, 1, 1, 0, 47, 5, 1, 21, 19, 0, 12, 0, 98, 1, 1, 1, 59, 0, 1, 1, 67, 0, 14, 0, 27, 22, 1, 0, 151, 7, 21, 1, 31, 0, 76, 1, 87, 1, 1, 0, 82, 0, 1, 28, 258, 1, 18, 0, 39, 1, 16, 0, 203, 0, 1, 26, 43, 1, 20, 0, 219, 102, 1, 0, 104, 1, 1, 1, 127, 0, 99, 1, 51, 1, 1, 1, 423

Offset: 1

Author

Antti Karttunen, May 24 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000203, A001065, A003415, A168036, A211991, A319683, A319684.

Inverse Möbius transform of A344178.

Programs

Mathematica

Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] - DivisorSigma[1, #] + # &, 96]] (* Michael De Vlieger, May 24 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A319684(n) = sumdiv(n, d, A003415(d));
A344584(n) = (A319684(n) - (sigma(n)-n));

Formula

a(n) = A319684(n) - A001065(n) = A211991(n) + A319683(n).

a(n) = Sum_{d|n} A344178(d).

a(n) = n + Sum_{d|n} A168036(d).