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A048975
Pairs of consecutive primes p, q, for which the smallest primitive root of q is 1 greater than the smallest primitive root of p.
1
2, 3, 5, 7, 13, 17, 29, 31, 83, 89, 131, 137, 197, 199, 211, 223, 317, 331, 349, 353, 443, 449, 461, 463, 509, 521, 563, 569, 587, 593, 613, 617, 619, 631, 727, 733, 797, 809, 821, 823, 853, 857, 877, 881, 947, 953, 967, 971, 983, 991, 991, 997, 1061, 1063
OFFSET
1,1
REFERENCES
Paulo Ribenboim, The new book of prime number records, Springer, 1996, pp. 22-25.
EXAMPLE
The primitive roots (mod 13) are 2, 6, 7, and 11, and the primitive roots (mod 17) are 3, 5, 6, 7, 10, 11, 12, and 14, so since 3 = 2 + 1, 13 and 17 are in the sequence.
991 is in the sequence twice because the smallest primitive roots for the three consecutive primes 983, 991, and 997 are 5, 6, and 7, respectively, so 991 appears as the larger of the pair (983,991) since 6 = 5 + 1, and as the smaller of the pair (991,997) since 7 = 6 + 1.
MATHEMATICA
Reap[ Do[ If[ PrimitiveRoot[p] + 1 == PrimitiveRoot[np = NextPrime[p]], Sow[p]; Sow[np]], {p, Prime /@ Range[200]}]][[2, 1]] (* Jean-François Alcover, Oct 04 2012 *)
Flatten[{Prime[#], Prime[#+1]}&/@Flatten[Position[Partition[ PrimitiveRoot[ Prime[ Range[200]]], 2, 1], _?(#[[2]]-#[[1]]==1&), {1}, Heads->False]]] (* Harvey P. Dale, Dec 22 2014 *)
PROG
(PARI) forprime(p=2, 1061, if( lift(znprimroot(p)) + 1 == lift(znprimroot(nextprime(p+1))), print(p); print(nextprime(p+1)))) \\ Michael B. Porter, Mar 03 2013
CROSSREFS
Sequence in context: A188809 A350443 A152449 * A009571 A087520 A117159
KEYWORD
nice,nonn
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Apr 22 2000
STATUS
approved