A328237 Quotient A003415(n+k)/A003415(n) for the least k > 0 for which such a quotient is an integer. Here A003415(x) is the arithmetic derivative of x.

1, 4, 3, 5, 2, 12, 2, 4, 3, 16, 2, 9, 3, 4, 1, 21, 1, 24, 2, 1, 3, 44, 4, 8, 4, 3, 6, 31, 1, 80, 1, 8, 8, 5, 4, 21, 3, 3, 4, 41, 3, 48, 4, 4, 3, 112, 4, 4, 1, 4, 8, 81, 2, 12, 1, 8, 4, 92, 4, 33, 5, 9, 2, 4, 1, 72, 3, 6, 4, 156, 4, 39, 5, 1, 3, 6, 1, 176, 1, 2, 3, 124, 2, 1, 12, 3, 1, 123, 1, 7, 4, 8, 3, 9, 1, 77, 3, 1, 4

Offset: 2

Author

Antti Karttunen, Oct 08 2019

Keywords

• nonn

Examples

			Arithmetic derivative of 6 is A003415(6) = 5. Not until at k=21 we find another number whose arithmetic derivative is a multiple of five (as A003415(21) = 10 = 2*5), thus a(6) = 10/5 = 2.
		

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 (gives the corresponding k).

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328237(n) = { my(d=A003415(n),t); for(k=1,oo,if(!((t=A003415(n+k))%d), return(t/d))); };

Formula

a(n) = A003415(n+A328235(n)) / A003415(n).