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.

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for primes by primitive root

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

Adjacent sequences: A048972 A048973 A048974 * A048976 A048977 A048978

Keyword

nice,nonn

Author

Felice Russo

Extensions

More terms from James A. Sellers, Apr 22 2000

Status

approved