A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 6, 1, 6, 4, 1, 1, 8, 1, 12, 3, 10, 1, 6, 1, 12, 8, 2, 1, 8, 1, 15, 5, 16, 12, 24, 1, 18, 12, 4, 1, 12, 1, 30, 8, 22, 1, 30, 1, 24, 16, 12, 1, 16, 20, 18, 9, 28, 1, 24, 1, 30, 24, 5, 3, 4, 1, 48, 11, 8, 1, 24, 1, 36, 24, 18, 15, 24, 1, 60, 1, 40, 1, 36, 16, 42, 28, 10, 1, 32, 4, 66

Offset

1,6

Comments

Conjecture: a(n) = 1 iff n = 1 or in A050376. This is an infinitary analog of Lehmer's totient conjecture from 1932.

For all i, j > 1: a(i) = a(j) => A302777(i) = A302777(j), if the above conjecture holds.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Wikipedia, Lehmer's totient problem

Formula

a(n) = A091732(n) / A340087(n) = A091732(n) / gcd(n-1, A091732(n)).

For all n >= 1, a(A084400(n)) = 1.

Prog

(PARI)
ispow2(n) = (n && !bitand(n, n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((d<n)&&A302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340088(n) = { my(x=A091732(n)); (x/gcd(n-1, x)); };

Crossrefs

Cf. A050376, A084400, A091732, A302777, A340087, A340089.

Cf. also A160595.

Sequence in context: A096234 A362554 A361441 * A087114 A303753 A088242

Adjacent sequences: A340085 A340086 A340087 * A340089 A340090 A340091

Keyword

nonn

Author

Antti Karttunen, Dec 31 2020

Status

approved