A328235 The least k > 0 such that the arithmetic derivative of n+k is a multiple of the arithmetic derivative of n.

1, 1, 4, 1, 15, 1, 12, 11, 8, 1, 4, 1, 13, 1, 12, 1, 20, 1, 24, 4, 23, 1, 56, 7, 10, 27, 36, 1, 28, 1, 44, 15, 114, 1, 76, 1, 84, 5, 56, 1, 48, 1, 20, 27, 53, 1, 80, 3, 36, 25, 76, 1, 81, 9, 4, 23, 26, 1, 116, 1, 64, 207, 80, 3, 52, 1, 40, 3, 82, 1, 232, 1, 205, 31, 36, 4, 27, 1, 92, 27, 88, 1, 160, 36, 130, 5, 12, 1, 81, 9, 52, 3

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) = 21-6 = 15.
		

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, A328236, A328237 (gives the quotient).

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328235(n) = { my(d=A003415(n)); for(k=1,oo,if(!(A003415(n+k)%d), return(k))); };