cp's OEIS Frontend

A382224 Numbers k such that every element with maximal order mod k is prime.

%I A382224 #28 Mar 23 2025 17:05:14
%S A382224 3,4,5,6,7,8,9,10,12,14,16,18,20,22,24,28,30,36,42,54,60,78
%N A382224 Numbers k such that every element with maximal order mod k is prime.
%C A382224 A generalization of A382220 to include k such that Z/kZ is noncyclic.
%C A382224 There are no more terms up to k = 10^7.
%C A382224 Is this sequence finite? For similar reasons to those stated in A382220, it becomes increasingly less likely that the set of all maximal order elements mod k will contain only primes as k increases.
%e A382224 Elements with maximal order mod a(n) for 1 <= n <= 22:
%e A382224   3  [2]
%e A382224   4  [3]
%e A382224   5  [2, 3]
%e A382224   6  [5]
%e A382224   7  [3, 5]
%e A382224   8  [3, 5, 7]
%e A382224   9  [2, 5]
%e A382224  10  [3, 7]
%e A382224  12  [5, 7, 11]
%e A382224  14  [3, 5]
%e A382224  16  [3, 5, 11, 13]
%e A382224  18  [5, 11]
%e A382224  20  [3, 7, 13, 17]
%e A382224  22  [7, 13, 17, 19]
%e A382224  24  [5, 7, 11, 13, 17, 19, 23]
%e A382224  28  [3, 5, 11, 17, 19, 23]
%e A382224  30  [7, 13, 17, 23]
%e A382224  36  [5, 7, 11, 23, 29, 31]
%e A382224  42  [5, 11, 17, 19, 23, 31]
%e A382224  54  [5, 11, 23, 29, 41, 47]
%e A382224  60  [7, 13, 17, 23, 37, 43, 47, 53]
%e A382224  78  [7, 11, 19, 37, 41, 59, 67, 71]
%t A382224 Select[Range[2,100],And@@PrimeQ@Select[Range[(n=#)-1],MultiplicativeOrder[#,n]==CarmichaelLambda[n]&]&] (* _Giorgos Kalogeropoulos_, Mar 23 2025 *)
%o A382224 (PARI) isok(n) = my(m=lcm(apply(f->(f[1]-1)*f[1]^(f[2]-1-(f[1]==2&&f[2]>2)), Vec(factor(n)~)))); for(k=1, n-1, if(gcd(k,n)==1 && znorder(Mod(k,n))==m && isprime(k)==0, return(0)); if(k==n-1, return(1)))
%Y A382224 Cf. A002322, A382220 (a subsequence).
%K A382224 nonn,more
%O A382224 1,1
%A A382224 _Miles Englezou_, Mar 19 2025