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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.

%I A328237 #8 Oct 09 2019 20:56:33
%S A328237 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,
%T A328237 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,
%U A328237 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
%N 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.
%H A328237 Antti Karttunen, <a href="/A328237/b328237.txt">Table of n, a(n) for n = 2..16385</a>
%H A328237 Antti Karttunen, <a href="/A328237/a328237.txt">Data supplement: n, a(n) computed for n = 2..65537</a>
%F A328237 a(n) = A003415(n+A328235(n)) / A003415(n).
%e A328237 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.
%o A328237 (PARI)
%o A328237 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328237 A328237(n) = { my(d=A003415(n),t); for(k=1,oo,if(!((t=A003415(n+k))%d), return(t/d))); };
%Y A328237 Cf. A003415, A328235 (gives the corresponding k).
%K A328237 nonn
%O A328237 2,2
%A A328237 _Antti Karttunen_, Oct 08 2019