A323384 Smallest number with exactly n divisors in Eisenstein integers.

1, 2, 3, 7, 9, 6, 27, 14, 12, 18, 243, 21, 729, 54, 36, 56, 6561, 60, 19683, 63, 108, 486, 177147, 42, 144, 1458, 147, 189, 4782969, 180, 14348907, 182, 972, 13122, 432, 84, 387420489, 39366, 2916, 126, 3486784401, 540, 10460353203, 1701, 441, 354294, 94143178827, 168, 1728, 720

Offset: 1

Author

Jianing Song, Jan 12 2019

Keywords

• nonn

Comments

a(n) is the smallest k such that A319442(k) = n.

Analog of A005179 and A302252 over the ring of Eisenstein integers. The divisors are counted up to association.

Examples

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
The divisors of 14 in Eisenstein integers are 1, 2, 2 + w, 2 + w', 7, 4 + 2*w, 4 + 2*w', 14 and there associations, and 14 is the smallest number having exactly 8 divisors in Eisenstein integers, so a(8) = 14.
The divisors of 21 in Eisenstein integers are 1, 2*w - 1, 3, 2 + w, 2 + w', 5 - w, 5 - w', 6 + 3*w, 6 + 3*w', 7, 14*w - 7, 21 and there associations, and 21 is the smallest number having exactly 12 divisors in Eisenstein integers, so a(12) = 21.
		

Crossrefs

Cf. A005179, A302252, A319442 (number of divisors of n in Eisenstein integers).

Programs

PARI

a(n) = if(isprime(n)&&n>2, 3^((n-1)/2), my(k=1); while(A319442(k)!=n, k++); k)

Formula

For primes p > 2, a(p) = 3^((p-1)/2).