OFFSET
1,2
COMMENTS
Numbers k such that both k and k + 1 are in A033948.
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.
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.
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1001 from Michel Marcus)
EXAMPLE
5 is a primitive root modulo both 46 and 47, so 46 is a term.
2 is a primitive root modulo 53 and 5 is a primitive root modulo 54, so 53 is a term.
MATHEMATICA
q[n_] := q[n] = EulerPhi[n] == CarmichaelLambda[n]; Select[Range[720], q[#] && q[# + 1] &] (* Amiram Eldar, Jul 21 2024 *)
PROG
(PARI) isA033948(n) = (#znstar(n)[2]<=1)
isA319450(n) = isA033948(n)&&isA033948(n+1)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 19 2018
STATUS
approved