A302252 Smallest number with exactly n divisors in Gaussian integers.

1, 3, 2, 5, 4, 6, 8, 15, 16, 12, 32, 10, 64, 24, 36, 65, 256, 48, 512, 20, 72, 96, 2048, 30, 324, 192, 50, 40, 16384, 252, 32768, 195, 288, 768, 648, 80, 262144, 1536, 576, 60, 1048576, 504, 2097152, 160, 100, 6144, 8388608, 130, 5832, 1875, 2304, 320

Offset: 1

Author

Jianing Song, Apr 04 2018

Keywords

• nonn

Comments

The divisors are counted up to association.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Index entries for Gaussian integers and primes.

Crossrefs

Cf. A005179, A062327.

Programs

Mathematica

a[n_] := If[n > 2 && PrimeQ[n], 2^((n-1)/2), Block[{k=1}, While[ DivisorSigma[0, k, GaussianIntegers -> True] != n, k++]; k]]; Array[a, 52] (* Giovanni Resta, Apr 04 2018 *)

PARI

nbd(n) = {my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); if(p==2, r*=(2*e+1)); if(p%4==1, r*=(e+1)^2); if(p%4==3, r*=(e+1));); return(r);}  \\ A062327
a(n) = {my(k=1); while (nbd(k) != n, k++); k;} \\ Michel Marcus, Apr 26 2018

Formula

For prime p > 2, a(p) = 2^((p-1)/2) = sqrt(A005179(p)).

Extensions

More terms from Giovanni Resta, Apr 04 2018