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A351954 Arithmetic derivative without its inherited divisor applied to the prime shadow of the factorial base exp-function: a(n) = A342001(A181819(A276076(n))).

%I A351954 #19 Apr 30 2022 22:51:09
%S A351954 0,1,1,2,1,5,1,2,2,3,5,8,1,5,5,8,2,7,1,7,7,12,8,31,1,2,2,3,5,8,2,3,3,
%T A351954 4,8,11,5,8,8,11,7,10,7,12,12,17,31,46,1,5,5,8,2,7,5,8,8,11,7,10,2,7,
%U A351954 7,10,3,9,8,31,31,46,13,41,1,7,7,12,8,31,7,12,12,17,31,46,8,31,31,46,13,41,2,9,9,14,11
%N A351954 Arithmetic derivative without its inherited divisor applied to the prime shadow of the factorial base exp-function: a(n) = A342001(A181819(A276076(n))).
%C A351954 Compare the scatter plot to those of A275735, A353575 and of A353577. - _Antti Karttunen_, Apr 30 2022
%H A351954 Antti Karttunen, <a href="/A351954/b351954.txt">Table of n, a(n) for n = 0..20160</a>
%H A351954 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A351954 a(n) = A342001(A275735(n)) = A351945(A276076(n)).
%F A351954 a(n) = A353577(A351576(n)). - _Antti Karttunen_, Apr 30 2022
%o A351954 (PARI)
%o A351954 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A351954 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A351954 A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
%o A351954 A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
%o A351954 A342001(n) = (A003415(n) / A003557(n));
%o A351954 A351945(n) = A342001(A181819(n));
%o A351954 A351954(n) = A351945(A276076(n));
%Y A351954 Cf. A003415, A003557, A181819, A275735, A276076, A342001, A351576, A351945, A351952, A353575, A353577.
%Y A351954 Cf. A051683 (positions of 1's).
%K A351954 nonn,base,easy,look
%O A351954 0,4
%A A351954 _Antti Karttunen_, Apr 02 2022
%E A351954 Verbal description added to the definition by _Antti Karttunen_, Apr 30 2022