A368217 a(n) is the first number == 1 (mod n) that is the product of n primes, counted by multiplicity.

2, 9, 28, 81, 176, 15625, 288, 6561, 1792, 137781, 17920, 244140625, 30720, 7971615, 311296, 43046721, 1492992, 3814697265625, 2752512, 3486784401, 38797312, 242137805625, 28311552, 59604644775390625, 184549376, 51684605176023, 2583691264, 63546645708225, 9512681472, 41858774825571336448888891

Offset: 1

Author

Robert Israel, Dec 17 2023

Keywords

• nonn

Comments

a(n) is the first number k == 1 (mod n) such that A001222(k) = n.

A053669(n)^n <= a(n) <= A034694(n).

If n is in A007694 then a(n) = A053669(n)^n.

Examples

			a(4) = 81 because 81 == 1 (mod 4) and 81 = 3^4 is the product of 4 primes, counted by multiplicity, and no smaller number works.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001222, A007694, A034694, A053669.

Programs

Maple

f:= proc(n) uses priqueue; local p, x, Aprimes, v;
    initialize(Aprimes);
    p:= 2;
    while n mod p = 0 do p:= nextprime(p) od:
    insert([-p^n,p,0],Aprimes);
    do
      v:= extract(Aprimes);
      x:= -v[1];
      if x mod n = 1 then return x fi;
      if v[3] < n then
        insert([v[1],v[2],v[3]+1],Aprimes);
        p:= nextprime(v[2]);
        while n mod p = 0 do p:= nextprime(p) od;
        x:= x * (p/v[2])^(n-v[3]);
        insert([-x,p,v[3]],Aprimes);
      fi;
    od;
end proc:
f(1):= 2:
map(f, [$1..30]);