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A328307 a(n) tells how many numbers m there are in range 0..n such that the k-th arithmetic derivative of A276086(m) is zero for some k >= 0.

%I A328307 #12 Oct 12 2019 19:07:15
%S A328307 1,2,3,4,5,6,7,8,8,9,9,9,10,10,10,11,11,11,11,11,12,13,13,13,13,13,13,
%T A328307 13,14,14,15,16,17,18,18,19,19,20,21,21,22,22,22,23,23,23,24,25,25,26,
%U A328307 27,27,27,27,27,27,27,27,27,27,28,29,29,29,29,30,30,31,32,32,32,33,33,34,35,35,36,36,36,37,37,37,37,37,38
%N A328307 a(n) tells how many numbers m there are in range 0..n such that the k-th arithmetic derivative of A276086(m) is zero for some k >= 0.
%C A328307 a(n) = the number of such terms m in A328116 that m <= n.
%C A328307 Although in principle A276086 moves any n out of the "all hope lost" zone A100716 (where A328308 is always zero), back to its complement A048103, by comparing the ratio of this and A328309 it can be seen (see the Plot2-link in the Links-section) that such a transfer actually lessens the chances that by just iterating A003415 one could reach zero from there. Note also how the effect of the primorial base representation can be clearly seen in the folds and warps of that plot.
%H A328307 Antti Karttunen, <a href="/A328307/b328307.txt">Table of n, a(n) for n = 0..65537</a>
%H A328307 <a href="https://oeis.org/plot2a?name1=A328309&amp;name2=A328307&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Ratio of A328309(n)/A328307(n) plotted with OEIS Plot2-script</a>
%F A328307 a(0) = 1; for n > 0, a(n) = a(n-1) + A328306(n).
%F A328307 For all n >= 0, a(A328116(n)) = n.
%o A328307 (PARI)
%o A328307 up_to = 65537;
%o A328307 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 A328307 A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
%o A328307 A328308(n) = if(!n,1, while(n>1, n = A003415checked(n)); (n));
%o A328307 A328306(n) = A328308(A276086(n));
%o A328307 A328307list(up_to) = { my(v=vector(up_to), s=A328306(0)); for(i=1,up_to,s += A328306(i); v[i] = s); (v); };
%o A328307 v328307 = A328307list(up_to);
%o A328307 A328307(n) = if(!n,1,v328307[n]);
%Y A328307 Partial sums of A328306, a left inverse of A328116.
%Y A328307 Cf. A003415, A048103, A100716, A276086, A327969, A328306, A328308, A328309.
%K A328307 nonn
%O A328307 0,2
%A A328307 _Antti Karttunen_, Oct 12 2019