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

1,6

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

Formula

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

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

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 *)

Prog

(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)); };

Crossrefs

Cf. A005117, A047994, A345937, A345939.

Cf. also A160595, A340088, A345948.

Sequence in context: A308210 A112331 A133910 * A066441 A300384 A252890

Adjacent sequences: A345935 A345936 A345937 * A345939 A345940 A345941

Keyword

nonn

Author

Antti Karttunen, Jun 29 2021

Status

approved