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A056798

Prime powers with even nonnegative exponents.

1, 4, 9, 16, 25, 49, 64, 81, 121, 169, 256, 289, 361, 529, 625, 729, 841, 961, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 16384

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OFFSET

1,2

COMMENTS

Also numbers whose geometric mean of divisors is an integer. - Ctibor O. Zizka, Sep 29 2008

This is just a special case. In fact, the numbers whose geometric mean of divisors is an integer are all the squares of integers (A000290). - Daniel Lignon, Nov 29 2014

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A025473(n)^(2*A025474(n)) = A000961(n)^2;

A001222(a(n)) mod 2 = 0;

A003415(a(n)) = A192083(n); A068346(a(n)) = A192084(n). - Reinhard Zumkeller, Jun 26 2011

Sum_{n>=2} 1/a(n) = A154945. - Amiram Eldar, Sep 21 2020

MATHEMATICA

Take[Union[Flatten[Table[Prime[n]^k, {n, 31}, {k, 0, 14, 2}]]], 45] (* Alonso del Arte, Jul 05 2011 *)

PROG

(PARI) is(n)=my(e=isprimepower(n)); if(e, e%2==0, n==1) \\ Charles R Greathouse IV, Sep 18 2015

(Python)

from sympy import primepi, integer_nthroot

def A056798(n):

if n==1: return 1

def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x, k)[0])for k in range(2, x.bit_length(), 2)))

kmin, kmax = 1, 2

while f(kmax) >= kmax:

kmax <<= 1

while True:

kmid = kmax+kmin>>1

if f(kmid) < kmid:

kmax = kmid

else:

kmin = kmid

if kmax-kmin <= 1:

break

return kmax # Chai Wah Wu, Aug 13 2024

CROSSREFS

Cf. A000290, A000961, A025475, A154945.

Sequence in context: A069560 A075494 A063735 * A036454 A115648 A082522

Adjacent sequences: A056795 A056796 A056797 * A056799 A056800 A056801

KEYWORD

nonn

AUTHOR

Labos Elemer, Aug 28 2000

STATUS

approved