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A138291 Number of primes of the form prime(n)+g, where g is a primitive root of prime(n).

%I A138291 #7 Feb 16 2025 08:33:07
%S A138291 1,1,1,0,3,1,3,1,2,4,2,2,4,3,2,7,10,3,3,3,4,6,10,7,6,11,7,12,7,9,6,10,
%T A138291 14,10,17,10,10,12,11,13,22,7,9,11,16,10,5,13,23,8,23,12,9,23,26,22,
%U A138291 25,13,12,14,13,19,12,18,14,32,17,18,30,22,32,21,20,14,17,28,30,19,19,21
%N A138291 Number of primes of the form prime(n)+g, where g is a primitive root of prime(n).
%C A138291 It appears that only a(4), corresponding to the prime 7, is zero.
%H A138291 T. D. Noe, <a href="/A138291/b138291.txt">Table of n, a(n) for n=1..2000</a>
%H A138291 Eric Weisstein, <a href="https://mathworld.wolfram.com/PrimitiveRoot.html">MathWorld: Primitive Root</a>
%e A138291 a(5)=3 because the primitive roots of 11 are 2, 6, 7 and 8. Adding these numbers to 11 produce three primes: 13, 17 and 19.
%t A138291 Join[{1}, Table[p=Prime[n]; g=Select[Range[2,p-1], MultiplicativeOrder[ #,p]==p-1&]; Length[Select[p+g, PrimeQ]], {n,2,2000}]]
%t A138291 Table[Count[p+PrimitiveRootList[p],_?PrimeQ],{p,Prime[Range[80]]}] (* _Harvey P. Dale_, Aug 03 2021 *)
%Y A138291 Cf. A047934, A060749 (triangle of primitive roots of primes).
%K A138291 nonn
%O A138291 1,5
%A A138291 _T. D. Noe_, Mar 12 2008