cp's OEIS Frontend

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

%I A328235 #9 Oct 09 2019 20:56:24
%S A328235 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,
%T A328235 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,
%U A328235 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
%N A328235 The least k > 0 such that the arithmetic derivative of n+k is a multiple of the arithmetic derivative of n.
%H A328235 Antti Karttunen, <a href="/A328235/b328235.txt">Table of n, a(n) for n = 2..16385</a>
%H A328235 Antti Karttunen, <a href="/A328235/a328235.txt">Data supplement: n, a(n) computed for n = 2..65537</a>
%e A328235 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.
%o A328235 (PARI)
%o A328235 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328235 A328235(n) = { my(d=A003415(n)); for(k=1,oo,if(!(A003415(n+k)%d), return(k))); };
%Y A328235 Cf. A003415, A328236, A328237 (gives the quotient).
%K A328235 nonn
%O A328235 2,3
%A A328235 _Antti Karttunen_, Oct 08 2019