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A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

%I A340072 #16 Jan 04 2022 04:08:12
%S A340072 1,1,1,3,1,4,1,9,5,3,1,6,1,5,12,27,1,20,1,18,20,12,1,36,7,16,25,30,1,
%T A340072 6,1,81,3,9,15,15,1,11,16,27,1,20,1,18,20,28,1,54,11,42,36,12,1,100,4,
%U A340072 45,44,15,1,72,1,36,100,243,48,48,1,54,7,12,1,180,1,40,42,66,60,64,1,162,125,21,1,120,9,23,60,108
%N A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.
%C A340072 Prime shifted analog of A160595.
%H A340072 Antti Karttunen, <a href="/A340072/b340072.txt">Table of n, a(n) for n = 1..8191</a>
%H A340072 Antti Karttunen, <a href="/A340072/a340072.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A340072 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A340072 a(n) = A160595(A003961(n)).
%F A340072 a(n) = A003972(n) / A340071(n).
%p A340072 f:= proc(n) local F,x,p,t;
%p A340072   F:= ifactors(n)[2];
%p A340072   x:= mul(nextprime(t[1])^t[2],t=F);
%p A340072   p:= numtheory:-phi(x);
%p A340072   p/igcd(x-1,p)
%p A340072 end proc:
%p A340072 map(f,[$1..100]); # _Robert Israel_, Dec 28 2020
%t A340072 a[n_] := Module[{x, p, e, phi}, x = Product[{p, e} = pe; NextPrime[p]^e, {pe, FactorInteger[n]}]; phi = EulerPhi[x]; phi/GCD[x-1, phi]];
%t A340072 Array[a, 100] (* _Jean-François Alcover_, Jan 04 2022 *)
%o A340072 (PARI)
%o A340072 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A340072 A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };
%Y A340072 Cf. A000010, A003961, A003972, A160595, A253885, A340071, A340073, A340075 (gives the odd part).
%Y A340072 Cf. also A340082.
%K A340072 nonn
%O A340072 1,4
%A A340072 _Antti Karttunen_, Dec 28 2020