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A328309 a(n) counts the numbers in 0..n whose k-th arithmetic derivative is zero for some k >= 0.

%I A328309 #14 Jun 25 2022 12:03:05
%S A328309 1,2,3,4,4,5,6,7,7,8,9,10,10,11,12,12,12,13,14,15,15,16,17,18,18,19,
%T A328309 19,19,19,20,21,22,22,23,24,24,24,25,26,26,26,27,28,29,29,29,30,31,31,
%U A328309 32,32,32,32,33,33,33,33,34,35,36,36,37,38,38,38,39,40,41,41,41,42,43,43,44,44,44,44,45,46,47,47,47,48
%N A328309 a(n) counts the numbers in 0..n whose k-th arithmetic derivative is zero for some k >= 0.
%C A328309 a(n) is the number of terms m in A099308 such that m <= n.
%H A328309 Antti Karttunen, <a href="/A328309/b328309.txt">Table of n, a(n) for n = 0..65537</a>
%F A328309 a(0) = 1; for n > 0, a(n) = a(n-1) + A328308(n).
%F A328309 For all n >= 0, a(A099308(n)) = n.
%o A328309 (PARI)
%o A328309 up_to = 65537;
%o A328309 A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
%o A328309 A328308(n) = if(!n,1, while(n>1, n = A003415checked(n)); (n));
%o A328309 A328309list(up_to) = { my(v=vector(up_to), s=A328308(0)); for(i=1,up_to,s += A328308(i); v[i] = s); (v); };
%o A328309 v328309 = A328309list(up_to);
%o A328309 A328309(n) = if(!n,1,v328309[n]);
%Y A328309 Cf. A003415.
%Y A328309 Partial sums of A328308. A left inverse of A099308. Cf. also A328307.
%K A328309 nonn
%O A328309 0,2
%A A328309 _Antti Karttunen_, Oct 12 2019