A239628 Factored over the Gaussian integers, the least positive number having n prime factors counted multiply, including units -1, i, and -i.

1, 9, 2, 6, 4, 12, 8, 16, 48, 144, 32, 96, 64, 192, 128, 256, 768, 2304, 512, 1536, 1024, 3072, 2048, 4096, 12288, 36864, 8192, 24576, 16384, 49152, 32768, 65536, 196608, 589824, 131072, 393216, 262144, 786432, 524288, 1048576, 3145728, 9437184, 2097152

Offset: 1

Author

T. D. Noe, Mar 31 2014

Keywords

• nonn

Comments

Here -1, i, and -i are counted as factors. The factor 1 is counted only in a(1). All these numbers of products of 2^k, 3, and 9.

Similar to A164073, which gives the least integer having n prime factors (over the Gaussian integers) shifted by 1.

Examples

			a(2) = 9 because 9 = 3 * 3.
a(3) = 2 because 2 = -i * (1 + i)^2.
a(4) = 6 because 6 = -i * (1 + i)^2 * 3.
		

Crossrefs

Cf. A001221, A001222 (integer factorizations).

Cf. A078458, A086275 (Gaussian factorizations).

Cf. A164073 (least number having n Gaussian factors, excluding units);

Cf. A239627 (number of Gaussian factors of n, including units).

Cf. A239629, A239630 (similar, but count distinct prime factors).

Programs

Mathematica

nn = 30; t = Table[0, {nn}]; n = 0; found = 0; While[found < nn, n++; cnt = Total[Transpose[FactorInteger[n, GaussianIntegers -> True]][[2]]]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; t