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Arithmetic Derivative (Posted on 2022-06-10) Difficulty: 4 of 5
Consider the following (simplified) definition of the "Arithmetic Derivative" for n in positive integers:
D(0) = D(1) = 0
D(prime) = 1
D(ab) = D(a)*b + D(b)*a

Examples:
D(7) = 1 because 7 is prime.
D(30) = D(5*6) = D(5)*6 + 5*D(6) = 1*6 + 5*D(2*3)
= 6 + 5*[D(2)*3+2*D(3)] = 6 + 5*5 = 31, so ...
D(30) = 31
D(58) = 31 (More than one integer can have the same Arithmetic Derivative.)

(1). Find n and D(n) (n up to 5 digits) such that D(n) is the largest.
(2). Find n and D(n) (n up to 5 digits and not prime) such that the ratio D(n)/n is the largest.
(3). Which 4-digit Palindrome is the Arithmetic Derivative of the most 4-digit positive integers, and list them.
(4). For what set of n is n = D(n)

No Solution Yet Submitted by Larry    
Rating: 5.0000 (1 votes)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: bits and piecesCharlie2022-06-10 15:50:51
bits and piecesLarry2022-06-10 15:05:44
Python version for 1 & 2Charlie2022-06-10 14:47:36
re(2): part (3) expanded to 5-digit n (continued)Charlie2022-06-10 11:10:09
re: part (3) expanded to 5-digit n (continued)Charlie2022-06-10 11:09:09
part (3) expanded to 5-digit nCharlie2022-06-10 11:07:23
Solutioncomputer solutionCharlie2022-06-10 11:02:48
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