This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377402 #15 Nov 18 2024 07:42:55 %S A377402 1,3,7,211,43891,300690391 %N A377402 Least k such that the ratio of the number of residues mod k coprime to k and the number of primitive roots mod k is greater than or equal to n for k such that at least one primitive root mod k exists. Equivalently, k such that floor(phi(k)/phi(phi(k)) is a record value for those k belonging to A033948. %C A377402 These are the numbers k with the least proportion of primitive roots mod k to residues mod k coprime to k, of all m < k. %C A377402 Let U(k) be the group of units of the ring Z/kZ, and Gen(k) the set of distinct single element generators of U(k). Then a(n) is equivalently those k for which floor(|U(k)|/|Gen(k)|) is a record value for those k with cyclic U(k). %C A377402 Some data: %C A377402 --------------------------------------------------------------------------------- %C A377402 n a(n) log(a(n)) phi(a(n)) max prime(r)#|phi(a(n)) r %C A377402 --------------------------------------------------------------------------------- %C A377402 1 1 0 1 0 0 %C A377402 2 3 1.10... 2 2 1 %C A377402 3 7 1.96... 2*3 6 2 %C A377402 4 211 5.35... 2*3*5*7 210 4 %C A377402 5 43891 10.69... 2*3*5*7*11*19 2310 5 %C A377402 6 300690391 19.52... 2*3*5*7*11*13*17*19*31 9699690 8 %C A377402 --------------------------------------------------------------------------------- %C A377402 For n > 1, is a(n) necessarily prime? Furthermore, is a(n) necessarily a prime such that phi(a(n)) is squarefree? Lastly, is every phi(a(n)) divisible by a maximal primorial prime(r)# of length r <= omega(phi(a(n))), such that if a maximal prime(s)# divides phi(a(m)) and n < m, then r < s? %C A377402 Based on the behavior of log(a(n)), we may expect a(7) to be found in the vicinity of floor(exp(40)) = 235385266837019985. %F A377402 Numbers k for which floor(A000010(A033948(k))/A046144(A033948(k)) is a record value. %e A377402 There are 2 residues mod 3 coprime to 3, and only 1 is a primitive root. 3 is the least k for which the floor of the ratio is 2, and so a(2) = 3. %e A377402 There are 210 residues mod 211 coprime to 211, and 48 are primitive roots. Floor(210/48) = 4, and 211 is the least k for which the floor of the ratio is 4, and so a(4) = 211. %o A377402 (PARI) %o A377402 S=[1]; for(n=1,100000, if(#znstar(n).cyc>1,next); f=eulerphi; if(floor(f(n)/f(f(n)))>floor(f(S[length(S)])/f(f(S[length(S)]))), S=concat(S,n))); print(S) %Y A377402 Cf. A000010, A010554, A033948, A046144. %K A377402 nonn,more %O A377402 1,2 %A A377402 _Miles Englezou_, Oct 26 2024