A328383 a(n) gives the number of iterations of x -> A003415(x) needed to reach the first number which is either a divisor or multiple of n, but not both at the same time. If no such number can ever be reached, a(n) is 0 (when either n is of the form p^p, or if the iteration would never stop). When the number reached is a divisor of n, a(n) is -1 * iteration count.

-1, -1, 0, -1, -2, -1, 2, -3, -2, -1, 9, -1, -4, 23, 1, -1, -4, -1, 5, -2, -2, -1, 2, -3, 24, 0, 18, -1, -2, -1, 6, -5, -2, 85, 7, -1, -4, 21, 10, -1, -2, -1, 35, 53, -4, -1, 2, -5, 44, 18, 34, -1, 2, 21, 4, -3, -2, -1, 16, -1, -6, 21, 1, -5, -2, -1, 7, 85, -2, -1, 4, -1, 23, 55, 5, -4, -2, -1, 4, 9, -2, -1, 42, -3, 42

Offset: 2

Author

Antti Karttunen, Oct 15 2019

Keywords

• sign
• hard

Comments

The absolute value of a(n) tells how many columns right from the leftmost column in array A258651 one needs to go at row n, before one (again) finds either a divisor or a multiple of n, with 0's reserved for cases like 4 and 27 where the same value continues forever. If one finds a divisor before a multiple, then the value of a(n) will be negative, otherwise it will be positive.

Question: What is the value of a(91) ?

Examples

			For n = 6, its arithmetic derivative A003415(6) = 5 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(5) = 1 is its divisor, thus a(6) = -2.
For n = 8, its arithmetic derivative A003415(8) = 12 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(12) = 16 is its multiple, thus a(8) = +2.
Numbers reached for n=2..28 (with positions of the form p^p are filled with the same p^p): 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 8592, 1, 1, 410267320320, 32, 1, 1, 1, 240, 7, 1, 1, 48, 1, 410267320320, 27, 9541095424. For example, we have a(12) = 9 and the 9th arithmetic derivative of 12 is A003415^(9)(12) = 8592 = 716*12.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..90

Crossrefs

Cf. A003415, A258651, A328235, A328236.

Cf. A051674 (indices of zeros provided for all n >= 2 either a divisor or multiple can be found).

Cf. A256750, A328248, A328384 for similar counts.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328383(n) = { my(u=A003415(n),k=1); if(u==n,return(0)); while((n%u) && (u%n), k++; u = A003415(u)); if(u%n,-k,k); };

Formula

a(A000040(n)) = -1.

a(A051674(n)) = 0.