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A307711
a(n) is the least number k such that exactly fraction 1/n of the members of the reduced residue system mod k are prime, or 0 if there is no such k.
2
3, 31, 97, 331, 1009, 3067, 11513, 27403, 64621, 185617, 480853, 1333951, 3524431, 9558361, 26080333, 70411483, 189961939
OFFSET
2,1
COMMENTS
a(n) is the least number k, if any exists, such that A000010(k)/A048865(k) = n.
a(n) = A307712(m) for the least m such that A307713(m)=n.
FORMULA
n*A048865(a(n)) = A000010(a(n)).
EXAMPLE
Of the 30 members of the reduced residue system mod 31, exactly one-third, namely 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, are prime. 31 is the least number with this property, so a(3) = 31.
MAPLE
f:= proc(n) uses numtheory;
phi(n)/(pi(n) - nops(factorset(n)));
end proc:
N:= 13: # to get a(2)..a(N)
R:= Array(2..N): count:= 0:
for k from 3 while count < N-1 do
v:= f(k);
if v::integer and v <= N and R[v] = 0 then
R[v]:= k;
count:= count+1;
fi
od:
convert(R, list);
MATHEMATICA
With[{s = Table[EulerPhi[n]/Count[Prime@ Range@ PrimePi@ n, _?(GCD[#, n] == 1 &)], {n, 3, 10^4}]}, Array[2 + FirstPosition[s, #][[1]] &, Max@ Select[s, IntegerQ] - 1, 2]] (* Michael De Vlieger, Apr 23 2019 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
J. M. Bergot and Robert Israel, Apr 23 2019
STATUS
approved