A355667 Least number phi(k) such that n * phi(k) < k, where phi is Euler's totient function.

1, 2, 8, 48, 5760, 36495360, 1854081073152000, 400440702414394285778534400000, 165062110921422523175104166476600499887194872217600000000

Offset

1,2

Comments

a(n) is the Euler totient of A091456(n). Useful for calculating a simple upper bound of the inverse phi function.

Links

Nico Mexis, Table of n, a(n) for n = 1..13

John Abbott and Nico Mexis, Cyclotomic Factors and LRS-Degeneracy, arXiv:2403.08751 [math.AC], 2024. See pp. 9, 24.

Formula

a(n) = A000010(A091456(n)).

a(n) = A000010(A091439(n+1)).

Example

If we know n=276, then all numbers k with phi(k)=n will satisfy k < 5*n since m=a(5) is the first number in the sequence with n < m.

Mathematica

a[n_] := Module[{k = 1, p = 1, e}, While[k <= n*(e = EulerPhi[k]), p = NextPrime[p]; k *= p]; e]; Array[a, 9] (* Amiram Eldar, Jul 13 2022 *)

Prog

(PARI) a(n)=my(k=1); forprime(p=2, , if(n*eulerphi(k)<k, return(eulerphi(k)), k*=p)
(CoCoA) // CoCoA 5
define a(n) k := 1; p := 1; while k <= n*EulerTotient(k) do p := NextPrime(p); k := k*p; endwhile; return EulerTotient(k); enddefine;

Crossrefs

Totients of A091456 or A091439.

Sequence in context: A356658 A279239 A279109 * A366239 A364592 A183070

Adjacent sequences: A355664 A355665 A355666 * A355668 A355669 A355670

Keyword

nonn

Author

Nico Mexis, Jul 13 2022

Status

approved