%I #53 Jun 23 2024 21:42:56
%S 1,2,1,1,1,1,5,1,3,6,1,1,1,1,3,2,1,1,1,1,3,10,1,1,1,10,15,3,1,1,1,1,1,
%T 14,1,6,5,1,21,2,1,1,1,1,3,3,25,1,7,14,15,10,7,1,1,2,1,2,1,1,1,1,3,3,
%U 3,18,1,1,3,2,1,1,1,1,3,5,5,18,1,1,1,6,1,1,1,2,15,2,35,1,1,2,3,2,49,6,1,1,7,15,35,1,7,1,1,1
%N Greatest common divisor of the arithmetic derivative and the primorial base exp-function: a(n) = gcd(A003415(n), A276086(n)).
%C Sequence contains only terms of A048103.
%C Proof that A046337 gives the positions of even terms: see _Charlie Neder_'s Feb 25 2019 comment in A235992 and recall that A276086 is never a multiple of 4, as it is a permutation of A048103, and furthermore it toggles the parity. See also comment in A327860. - _Antti Karttunen_, May 01 2022
%H Antti Karttunen, <a href="/A327858/b327858.txt">Table of n, a(n) for n = 0..65537</a>
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F a(n) = gcd(A003415(n), A276086(n)).
%F a(p) = 1 for all primes p.
%F a(n) = A276086(A351234(n)). - _Antti Karttunen_, May 01 2022
%F From _Antti Karttunen_, Dec 05 2022: (Start)
%F For n >= 2, a(n) = gcd(A003415(n), A328382(n)).
%F (End)
%F For n >= 2, a(n) = A358669(n) / A359423(n). For n >= 1, A356299(n) | a(n). - _Antti Karttunen_, Jan 09 2023
%F a(n) = gcd(A003415(n), A373849(n)) = gcd(A276086(n), A369971(n)) = A373843(A276086(n)). - _Antti Karttunen_, Jun 21 & 23 2024
%t Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12], f, g}, f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; g[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[GCD[f@ #, g@ #] &, 105]] (* _Michael De Vlieger_, Sep 30 2019 *)
%o (PARI)
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A327858(n) = gcd(A003415(n),A276086(n));
%Y Cf. A003415, A048103, A235992, A276086, A327859, A328382, A351234, A354348, A356299, A358669, A359423, A359589 (Dirichlet inverse of a(n)-1), A369971, A373849.
%Y Cf. A046337 (positions of even terms), A356311 (positions of 1's), A356310 (their characteristic function).
%Y Cf. also A085731, A324198, A328572 [= gcd(A276086(n), A327860(n))], A345000, A373145, A373843.
%K nonn,base,easy
%O 0,2
%A _Antti Karttunen_, Sep 30 2019
%E Verbal description added to the definition by _Antti Karttunen_, May 01 2022