A345938 a(n) = uphi(n) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jun 29 2021

Keywords

• nonn

Comments

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Wikipedia, Lehmer's totient problem

Crossrefs

Cf. A005117, A047994, A345937, A345939.

Cf. also A160595, A340088, A345948.

Programs

Mathematica

uphi[1]=1;uphi[n_]:=Times@@(#[[1]]^#[[2]]-1&/@FactorInteger[n]);
a[n_]:=uphi[n]/GCD[n-1,uphi[n]];Array[a,100]  (* Giorgos Kalogeropoulos, Jun 30 2021 *)

PARI

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A345938(n) = { my(u=A047994(n)); (u/gcd(n-1, u)); };

Formula

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

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