A068328 Arithmetic derivatives of the squarefree numbers.

0, 1, 1, 1, 5, 1, 7, 1, 1, 9, 8, 1, 1, 10, 13, 1, 15, 1, 31, 1, 14, 19, 12, 1, 21, 16, 1, 41, 1, 25, 1, 20, 1, 16, 22, 31, 1, 1, 33, 18, 61, 1, 26, 59, 1, 1, 39, 18, 71, 1, 43, 1, 22, 45, 32, 1, 20, 34, 49, 24, 1, 1, 91, 1, 71, 55, 1, 1, 87, 40, 1, 101, 28, 61, 24, 63, 44, 1, 46

Offset: 1

Author

Reinhard Zumkeller, Feb 27 2002

Keywords

• nonn

Comments

a(n) and A005117(n) are coprime, cf. A085731. - Reinhard Zumkeller, May 10 2011

Examples

			a(65) = d(A005117(65)) = d(105) = d(3*35) = 3*d(35)+d(3)*35 = 3*d(5*7)+1*35 = 3*d(5*7)+1*35 = 3*(5*d(7)+d(5)*7)+35 = 3*(5*1+1*7)+35 = 3*12+35 = 71, where d(n) = A003415(n).
With d(1)=0, d(prime) = 1 and d(m*n) = d(m)*n + m*d(n).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6 (2003), Article 03.3.4.

Crossrefs

Cf. A003415, A005117, A069359, A085731.

Programs

Haskell

a068328 = a003415 . a005117 -- Reinhard Zumkeller, May 10 2011

Mathematica

ad[n_] := ad[n] = n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[1] = 0; Table[ad[k], {k, Select[Range[150], SquareFreeQ]}] (* Amiram Eldar, Mar 04 2024 *)

Formula

a(n) = A003415(A005117(n)).

a(n) = A069359(A005117(n)).

a(n) = Sum_{prime p | A005117(n)} A005117(n)/p.