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a(n) = gcd(A003415(n), A328572(n)), where A003415 is the arithmetic derivative and A328572 converts the primorial base expansion of n into its prime product form, but with 1 subtracted from all nonzero digits.
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%I #11 Feb 04 2022 16:09:26

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,5,5,1,1,1,1,1,1,1,

%T 1,3,1,1,1,1,1,1,1,1,1,1,5,1,1,1,5,5,1,1,1,1,1,1,1,1,1,1,1,1,3,3,1,1,

%U 1,1,1,1,1,1,1,5,5,3,1,1,1,1,1,1,1,1,5,1,35,1,1,1,1,1,49,3,1,1,7,1,7,1,7

%N a(n) = gcd(A003415(n), A328572(n)), where A003415 is the arithmetic derivative and A328572 converts the primorial base expansion of n into its prime product form, but with 1 subtracted from all nonzero digits.

%H Antti Karttunen, <a href="/A351086/b351086.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), A328572(n)).

%F a(n) = gcd(A327858(n), A345000(n)).

%o (PARI)

%o A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415

%o A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };

%o A351086(n) = gcd(A003415(n), A328572(n));

%o (PARI)

%o A351086(n) = { my(m=1, p=2, orgn=A003415(n)); while(n, if(n%p, m *= (p^min((n%p)-1, valuation(orgn, p)))); n = n\p; p = nextprime(1+p)); (m); };

%Y Cf. A003415, A276086, A327858, A345000, A351084, A351085.

%K nonn,easy

%O 0,22

%A _Antti Karttunen_, Feb 03 2022