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

%I A319048 #45 Dec 06 2024 16:30:49
%S A319048 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,
%T A319048 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,
%U A319048 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
%N A319048 a(n) is the greatest k such that A000010(k) divides n where A000010 is the Euler totient function.
%C A319048 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).
%H A319048 Antti Karttunen, <a href="/A319048/b319048.txt">Table of n, a(n) for n = 1..16384</a>
%H A319048 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A319048 József Sándor, <a href="http://nntdm.net/volume-15-2009/number-3/01-08/">On the Euler minimum and maximum functions</a>, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, Pages 1—8.
%F A319048 a(n) = Max_{d|n} A057635(d). - _Antti Karttunen_, Sep 09 2018
%t A319048 a[n_] := Module[{kmax = If[n <= 6, 10 n, n^2]}, For[k = kmax, True, k--, If[Divisible[n, EulerPhi[k]], Return[k]]]];
%t A319048 Array[a, 80] (* _Jean-François Alcover_, Sep 17 2018, from PARI *)
%o A319048 (PARI) a(n) = {my(kmax = if (n<=6, 10*n, n^2)); forstep (k=kmax, 1, -1, if ((n % eulerphi(k)) == 0, return (k)););}
%o A319048 (PARI)
%o A319048 \\ (The first two functions could probably be combined in a smarter way):
%o A319048 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_
%o A319048 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))));
%o A319048 A319048(n) = { my(m=0); fordiv(n,d, m = max(m,A057635(d))); (m); }; \\ _Antti Karttunen_, Sep 09 2018
%o A319048 (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
%Y A319048 Cf. A000010 (Euler totient), A061026 (the totient minimum function).
%Y A319048 Cf. A319068 (the analog for the sum of divisors).
%Y A319048 Cf. A070633, also A057635, A057826, A069932, A071181.
%Y A319048 Right border of A378638.
%K A319048 nonn
%O A319048 1,1
%A A319048 _Michel Marcus_, Sep 09 2018