A377940 a(n) is the least number k such that the least j > k for which k is a primitive root is also the least j > n + k for which n + k is a primitive root, or -1 if there is no such k.

20, 6, 32, 10, 39, 28, 38, 18, 18, 42, 46, 42, 88, 42, 46, 173, 46, 78, 229, 102, 102, 294, 78, 150, 210, 150, 210, 193, 232, 193, 848, 488, 330, 226, 226, 328, 488, 328, 294, 172, 172, 294, 294, 294, 462, 328, 736, 328, 328, 294, 1098, 328, 328, 1196, 172, 172, 1322, 172, 1196, 856, 1108, 889

Offset: 1

Author

Robert Israel, Nov 11 2024

Keywords

• nonn

Comments

a(n) is the least k, if it exists, such that 0 < A377938(k) = A377938(k+n).

Examples

			a(3) = 32 because A377938(32) = A377938(35) = 37, i.e. 37 is the least number j > 32 such that 32 is a primitive root mod j and the least number j > 35 such that 35 is a primitive root mod j, and no number less than 32 works.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A377938.

Programs

Maple

N:= 10^6:
P:= select(isprime, {seq(i,i=3..N,2)}):Cands:= map(proc(t) local i; (seq(t^i,i=1..ilog[t](N)), seq(2*t^i,i=1..ilog[t](N/2))) end proc,P):
Cands:= sort(convert({4} union Cands, list)):
nC:= nops(Cands):
Phis:= map(numtheory:-phi, Cands):
f:= proc(n)
option remember;
local k0,k;
      if issqr(n) then return -1 fi;
      k0:= ListTools:-BinaryPlace(Cands,n)+1;
      for k from k0 to nC do
        if igcd(Cands[k],n) = 1 and numtheory:-order(n,Cands[k]) = Phis[k] then return Cands[k] fi
      od;
    FAIL
end proc:
g:= proc(n)
  local k;
  for k from n+1 do
    if f(k) > 0 and f(k) = f(k+n) then return k
  elif f(k) = FAIL and f(k+n) = FAIL then return FAIL fi
  od
end proc:
map(g, [$1..200]);