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

1,5

Comments

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

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.

Formula

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

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

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

Example

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).

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 *)

Prog

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

Crossrefs

Cf. A003415, A005117, A069359, A085731.

Sequence in context: A366284 A364491 A190643 * A342918 A109375 A051712

Adjacent sequences: A068325 A068326 A068327 * A068329 A068330 A068331

Keyword

nonn

Author

Reinhard Zumkeller, Feb 27 2002

Status

approved