A345938 - OEIS

%I #17 Jul 04 2021 15:54:13

%S 1,1,1,1,1,2,1,1,1,4,1,6,1,6,4,1,1,8,1,12,3,10,1,14,1,12,1,2,1,8,1,1,

%T 5,16,12,24,1,18,12,28,1,12,1,30,8,22,1,30,1,24,16,12,1,26,20,42,9,28,

%U 1,24,1,30,24,1,3,4,1,48,11,8,1,56,1,36,24,18,15,24,1,60,1,40,1,36,16,42,28,70,1,32,4,66,15

%N a(n) = uphi(n) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

%C For all squarefree n (A005117), a(n) = A160595(n), thus if there are any composite solutions to the Lehmer's totient conjecture, then they give also a such a subset of positions of 1's in this sequence that are not powers of primes. See comments in A160595.

%H Antti Karttunen, <a href="/A345938/b345938.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A345938/a345938.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer&#39;s_totient_problem">Lehmer's totient problem</a>

%F a(n) = A047994(n) / A345937(n) = A047994(n) / gcd(n-1, A047994(n)).

%F a(2n-1) = A345948(2n-1), for all n >= 1.

%t uphi[1]=1;uphi[n_]:=Times@@(#[[1]]^#[[2]]-1&/@FactorInteger[n]);

%t a[n_]:=uphi[n]/GCD[n-1,uphi[n]];Array[a,100]  (* _Giorgos Kalogeropoulos_, Jun 30 2021 *)

%o (PARI)

%o A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };

%o A345938(n) = { my(u=A047994(n)); (u/gcd(n-1, u)); };

%Y Cf. A005117, A047994, A345937, A345939.

%Y Cf. also A160595, A340088, A345948.

%K nonn

%O 1,6

%A _Antti Karttunen_, Jun 29 2021