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A319450 Numbers k such that k and k + 1 both have primitive roots.

%I A319450 #20 Jul 21 2024 03:45:29
%S A319450 1,2,3,4,5,6,9,10,13,17,18,22,25,26,37,46,49,53,58,61,73,81,82,97,106,
%T A319450 121,157,162,166,178,193,226,241,242,250,262,277,313,337,346,358,361,
%U A319450 382,397,421,457,466,478,486,502,541,562,577,586,613,625,661,673,718
%N A319450 Numbers k such that k and k + 1 both have primitive roots.
%C A319450 Numbers k such that both k and k + 1 are in A033948.
%C A319450 Apart from the first four terms, numbers k such that there exists odd primes p, q and positive numbers u, v such that k = p^u, k + 1 = 2*q^v or k = 2*p^u, k + 1 = q^v.
%C A319450 Let p be an odd prime. If 2*p^e - 1 is prime, then 2*p^e - 1 is a term. If 2*p^e + 1 is prime, then 2*p^e is a term. If (p^2^e + 1)/2 is prime, then p^2^e is a term. However it's not known whether there are infinitely many primes of the form 2*p^e +- 1 or (p^2^e + 1)/2.
%C A319450 The case that k and k + 1 are both in this sequence is extremely rare. Only 11 such k are known: 1, 2, 3, 4, 5, 9, 17, 25, 81, 241 and 3^541 - 2. It's possible that there are no further members. See A305237.
%H A319450 Amiram Eldar, <a href="/A319450/b319450.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1001 from Michel Marcus)
%e A319450 5 is a primitive root modulo both 46 and 47, so 46 is a term.
%e A319450 2 is a primitive root modulo 53 and 5 is a primitive root modulo 54, so 53 is a term.
%t A319450 q[n_] := q[n] = EulerPhi[n] == CarmichaelLambda[n]; Select[Range[720],q[#] && q[# + 1] &] (* _Amiram Eldar_, Jul 21 2024 *)
%o A319450 (PARI) isA033948(n) = (#znstar(n)[2]<=1)
%o A319450 isA319450(n) = isA033948(n)&&isA033948(n+1)
%Y A319450 Cf. A033948, A305237.
%K A319450 nonn
%O A319450 1,2
%A A319450 _Jianing Song_, Sep 19 2018