A225239 Numbers n such that there is an integer k with the property that k^tau(n) = sigma(n).

1, 3, 217, 862, 1177, 1207, 1219, 3937, 8743, 9481, 13822, 18137, 19567, 19849, 20057, 20257, 20299, 20437, 33607, 57337, 91847, 96217, 100579, 103897, 154969, 157921, 158623, 228889, 233047, 304117, 324817, 325579, 329057, 330529, 537817, 595417, 608287

Offset

1,2

Comments

Corresponding values of k: 1, 2, 4, 6, 6, 6, 6, 8, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 14, 16, 18, 18, 18, 18, 20, 20, 20, 22, 22, 24, 24, 24, 24, 24, 28, 28, 28, ... (see A225369).

Conjecture: all terms are squarefree numbers.

Conjecture is false: p = (312^6 / 13) - 1 = 70955197267967 is prime, so sigma(9*p) = sigma(9)*sigma(p) = 13*(p+1) = 312^6 = 312^tau(9*p). - Charlie Neder, Oct 05 2018

Links

Donovan Johnson, Table of n, a(n) for n = 1..500

Example

a(4) = 862 because sigma(862) = 1296 = 6^tau(862) = 6^4; k = 6.

Prog

(PARI) c=1; write("b225239.txt", c " " 1); for(n=2, 1943881801, s=sigma(n); if(ispower(s), nd=numdiv(n); r=round(sqrtn(s, nd)); if(r^nd==s, c++; write("b225239.txt", c " " n)))) /* Donovan Johnson, May 05 2013 */
(PARI) isok(n) = if (n==1, return(1)); my(s=sigma(n)); if(ispower(s), my(nd=numdiv(n)); r=sqrtnint(s, nd); (r^nd==s); ); \\ Michel Marcus, Feb 19 2020

Crossrefs

Cf. A000005 (tau(n): number of divisors of n).

Cf. A000203 (sigma(n): sum of divisors of n).

Cf. A051281 (sigma(n) is a power of tau(n)), A225369.

Sequence in context: A299225 A300037 A342234 * A225362 A063836 A172811

Adjacent sequences: A225236 A225237 A225238 * A225240 A225241 A225242

Keyword

nonn

Author

Jaroslav Krizek, May 04 2013

Status

approved