A328238 Quotient A003415(n*m)/A003415(n) for the least m > 1 for which such a quotient is an integer. Here A003415(x) is the arithmetic derivative of x.

4, 5, 3, 7, 12, 9, 16, 10, 20, 13, 7, 15, 28, 42, 6, 19, 16, 21, 18, 42, 44, 25, 48, 14, 52, 3, 26, 31, 60, 33, 8, 66, 68, 43, 9, 39, 56, 51, 80, 43, 84, 45, 19, 37, 92, 49, 13, 18, 24, 236, 46, 55, 16, 67, 112, 138, 88, 61, 116, 63, 124, 124, 12, 130, 132, 69, 82, 138, 140, 73, 28, 75, 119, 84, 108, 142, 156, 81, 44, 7, 164

Offset: 2

Author

Antti Karttunen, Oct 08 2019

Keywords

• nonn

Examples

			Arithmetic derivative of 6 is 6' = A003415(6) = 5. Taking arithmetic derivatives of its successive multiples, 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) = 60/5 = 12.
		

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, A328237.

Cf. A328236 (gives the corresponding m).

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328238(n) = { my(d=A003415(n),t); for(m=2,oo,if(!((t=A003415(n*m))%d), return(t/d))); };

Formula

a(n) = A003415(n*A328236(n)) / A003415(n).