A328236 The least m > 1 such that the arithmetic derivative of m*n is a multiple of the arithmetic derivative of n.

2, 2, 2, 2, 6, 2, 8, 4, 10, 2, 4, 2, 14, 12, 4, 2, 10, 2, 8, 21, 22, 2, 24, 4, 26, 2, 12, 2, 30, 2, 6, 33, 34, 8, 6, 2, 18, 12, 40, 2, 42, 2, 8, 22, 46, 2, 10, 4, 14, 32, 33, 2, 8, 12, 56, 24, 30, 2, 56, 2, 62, 40, 8, 65, 66, 2, 65, 69, 70, 2, 22, 2, 45, 24, 32, 65, 78, 2, 24, 4, 82, 2, 30, 24, 50, 16, 88, 2, 42, 32, 20, 40

Offset: 2

Author

Antti Karttunen, Oct 08 2019

Keywords

• nonn

Examples

			Arithmetic derivative of 6 is 6' = A003415(6) = 5. Taking arithmetic derivatives of successive multiples of six we obtain 12' = 16, 18' = 21, 24' = 44, 30' = 31, and not until with A003415(6*6) = 36' = 60 we obtain a multiple of 5. Thus a(6) = 6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..16385
• Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Crossrefs

Cf. A003415, A328235, A328238 (gives the corresponding quotients).

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328236(n) = { my(d=A003415(n)); for(m=2,oo,if(!(A003415(n*m)%d), return(m))); };