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

1,2

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.

Links

Table of n, a(n) for n=1..50.

Formula

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

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A053960 A114056 A168222 * A349641 A140221 A046668

Adjacent sequences: A323381 A323382 A323383 * A323385 A323386 A323387

Keyword

nonn

Author

Jianing Song, Jan 12 2019

Status

approved