A319048 a(n) is the greatest k such that A000010(k) divides n where A000010 is the Euler totient function.

2, 6, 2, 12, 2, 18, 2, 30, 2, 22, 2, 42, 2, 6, 2, 60, 2, 54, 2, 66, 2, 46, 2, 90, 2, 6, 2, 58, 2, 62, 2, 120, 2, 6, 2, 126, 2, 6, 2, 150, 2, 98, 2, 138, 2, 94, 2, 210, 2, 22, 2, 106, 2, 162, 2, 174, 2, 118, 2, 198, 2, 6, 2, 240, 2, 134, 2, 12, 2, 142, 2, 270, 2, 6, 2, 12, 2, 158, 2, 330

Offset

1,1

Comments

Sándor calls this function the totient maximum function and remarks that this function is well-defined, since a(n) can be at least 2, and cannot be greater than n^2 (when n > 6).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

József Sándor, On the Euler minimum and maximum functions, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, Pages 1—8.

Formula

a(n) = Max_{d|n} A057635(d). - Antti Karttunen, Sep 09 2018

Mathematica

a[n_] := Module[{kmax = If[n <= 6, 10 n, n^2]}, For[k = kmax, True, k--, If[Divisible[n, EulerPhi[k]], Return[k]]]];
Array[a, 80] (* Jean-François Alcover, Sep 17 2018, from PARI *)

Prog

(PARI) a(n) = {my(kmax = if (n<=6, 10*n, n^2)); forstep (k=kmax, 1, -1, if ((n % eulerphi(k)) == 0, return (k)); ); }
(PARI)
\\ (The first two functions could probably be combined in a smarter way):
A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ From A014197 by M. F. Hasler
A057635(n) = if(1==n, 2, if((n%2), 0, my(k=A014197(n), i=n); if(!k, 0, while(k, i++; if(eulerphi(i)==n, k--)); (i))));
A319048(n) = { my(m=0); fordiv(n, d, m = max(m, A057635(d))); (m); }; \\ Antti Karttunen, Sep 09 2018
(PARI) a(n) = {my(d = divisors(n)); vecmax(vector(#d, i, invphiMax(d[i]))); } \\ Amiram Eldar, Nov 29 2024, using Max Alekseyev's invphi.gp

Crossrefs

Cf. A000010 (Euler totient), A061026 (the totient minimum function).

Cf. A319068 (the analog for the sum of divisors).

Cf. A070633, also A057635, A057826, A069932, A071181.

Right border of A378638.

Sequence in context: A223473 A292522 A243197 * A066900 A356016 A336852

Adjacent sequences: A319045 A319046 A319047 * A319049 A319050 A319051

Keyword

nonn

Author

Michel Marcus, Sep 09 2018

Status

approved