cp's OEIS Frontend

A374034 a(n) = A276150(gcd(A276085(n), A328768(n))), where A276150 is the digit sum in primorial base, A276085 is the primorial base log-function, and A328768 is the first primorial based variant of arithmetic derivative.

%I A374034 #8 Jun 27 2024 10:20:46
%S A374034 0,1,1,1,1,1,1,2,2,1,1,2,1,1,2,2,1,1,1,2,2,1,1,1,2,1,1,2,1,1,1,3,2,1,
%T A374034 2,1,1,1,2,3,1,1,1,2,1,1,1,1,2,1,2,2,1,1,2,2,4,1,1,1,1,1,1,1,2,1,1,2,
%U A374034 2,1,1,1,1,1,1,2,4,1,1,1,2,1,1,1,2,1,2,2,1,1,6,2,2,1,2,1,1,1,1,1,1,1,1,2,3
%N A374034 a(n) = A276150(gcd(A276085(n), A328768(n))), where A276150 is the digit sum in primorial base, A276085 is the primorial base log-function, and A328768 is the first primorial based variant of arithmetic derivative.
%H A374034 Antti Karttunen, <a href="/A374034/b374034.txt">Table of n, a(n) for n = 1..100000</a>
%H A374034 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A374034 a(n) = A276150(A374031(n)).
%F A374034 Apparently, a(n) <= A328771(n) for all n >= 1.
%o A374034 (PARI)
%o A374034 A002110(n) = prod(i=1,n,prime(i));
%o A374034 A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
%o A374034 A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
%o A374034 A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
%o A374034 A374034(n) = A276150(gcd(A276085(n), A328768(n)));
%Y A374034 Cf. A002110, A276085, A276150, A328768, A374031.
%Y A374034 Cf. also A328771, A373989.
%K A374034 nonn
%O A374034 1,8
%A A374034 _Antti Karttunen_, Jun 27 2024