A219338 Numbers n for which n = (tau(n) - 1)^k with integer k.

4, 16, 27, 3125, 3375, 65536, 823543, 3748096, 52521875, 285311670611, 7625597484987, 302875106592253, 1156831381426176, 66182427701415936, 827240261886336764177, 511324276025564512546607, 1978419655660313589123979, 281633339785852578930098176

Offset: 1

Author

Zdenek Cervenka, Nov 18 2012

Keywords

• nonn

Comments

tau(n) is the number of positive divisors of n.

Examples

			a(1) = 4 because (tau(4) - 1)^2 = (3 - 1)^2 = 4 and this is the first number satisfying this condition.
a(2) = 16 because (tau(16) - 1)^2 = (5 - 1)^2 = 16 and this is the second number satisfying this condition.
a(3) = 27 because (tau(27) - 1)^3 = (4 - 1)^3 = 27 and this is the third number satisfying this condition.
		

Crossrefs

Cf. A180936.

Programs

Mathematica

Select[Range[10^4], IntegerQ[Log[DivisorSigma[0, #] - 1, #]] &] (* Alonso del Arte, Nov 18 2012 *)

PARI

v=vector(18); mx=3*10^26; c=0; for(m=2, 3440639, for(k=2, 87, n=m^k; if(n>mx, next(2)); if(m==numdiv(n)-1, c++; v[c]=n))); v=vecsort(v); for(i=1, c, print1(v[i]", ")) /* Donovan Johnson, Nov 19 2012 */

Formula

Numbers n for which n = (tau(n) - 1)^k with integer k.

Extensions

a(10)-a(18) from Donovan Johnson, Nov 19 2012