A275816 Least number k such that the number of its divisors is n times, with n>1, the number of its prime factors, counted without multiplicity.

2, 4, 8, 16, 32, 64, 128, 256, 432, 1024, 864, 4096, 1728, 2592, 3456, 65536, 6912, 262144, 10368, 14400, 27648, 4194304, 21600, 32400, 110592, 50400, 43200, 268435456, 64800, 1073741824, 86400, 230400, 1769472, 129600, 151200, 68719476736, 7077888, 921600

Offset

2,1

Comments

The number of divisors of p^(n-1) is n times the number of prime factors of p^(n-1), where p is prime. It means that a solution exists for every n>1.

Links

Giovanni Resta, Table of n, a(n) for n = 2..100

Formula

Least solution k of the equation A000005(k) = n * A001221(k).

Example

a(10) = 432 because the number of divisors of 432 is 20, the number of different prime factors of 432 is 2 (2, 3), and 20 = 10 * 2.

Maple

with(numtheory): P:=proc(q) local k, n; for n from 2 to q do for k from 1 to 2^(n-1) do
if tau(k)=n*nops(factorset(k)) then print(k); break; fi; od; od; end: P(10^9);

Mathematica

a[n_] := Block[{k = 2}, If[PrimeQ[n], 2^n/2, While[ DivisorSigma[0, k]/ PrimeNu[k] != n, k++]; k]]; a /@ Range[2, 25] (* Giovanni Resta, Nov 16 2016 *)

Crossrefs

Cf. A000005, A001221, A275819.

Sequence in context: A230579 A361937 A009694 * A097000 A285894 A271481

Adjacent sequences: A275813 A275814 A275815 * A275817 A275818 A275819

Keyword

nonn,easy

Author

Paolo P. Lava, Nov 15 2016

Extensions

a(29)-a(39) from Giovanni Resta, Nov 16 2016

Status

approved