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A370120 a(n) = A276085(A370117(n)), where A370117(n) is the denominator of n/A276086(A003415(n)), A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A276085 is its inverse, the primorial base log-function.

%I A370120 #7 Feb 11 2024 14:20:21
%S A370120 0,0,0,1,4,1,2,1,12,6,0,1,14,1,8,0,32,1,18,1,18,8,12,1,42,4,14,25,2,1,
%T A370120 30,1,80,12,18,6,60,1,20,14,62,1,8,1,48,31,24,1,110,14,32,18,56,1,78,
%U A370120 10,62,20,30,1,90,1,32,19,192,12,60,1,72,24,22,1,156,1,38,43,80,18,68,1,170,108,42,1,92,16,44
%N A370120 a(n) = A276085(A370117(n)), where A370117(n) is the denominator of n/A276086(A003415(n)), A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A276085 is its inverse, the primorial base log-function.
%H A370120 Antti Karttunen, <a href="/A370120/b370120.txt">Table of n, a(n) for n = 0..16384</a>
%F A370120 a(n) = A276085(A370117(n)).
%o A370120 (PARI)
%o A370120 A002110(n) = prod(i=1,n,prime(i));
%o A370120 A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
%o A370120 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A370120 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A370120 A370120(n) = { my(u=A276086(A003415(n))); A276085(u/gcd(n, u)); };
%Y A370120 Cf. A003415, A276085, A276086, A327859, A369964, A370115 (positions of 0's), A370117.
%K A370120 nonn
%O A370120 0,5
%A A370120 _Antti Karttunen_, Feb 11 2024