A192084 -id:A192084 - cp's OEIS frontend

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

Offset: 1

Labos Elemer, Aug 28 2000

nonn

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

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

Cf. A000290, A000961, A025475, A154945.

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

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

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

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

A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

Original entry on oeis.org

0, 4, 6, 32, 10, 14, 192, 108, 22, 26, 1024, 34, 38, 46, 500, 1458, 58, 62, 5120, 74, 82, 86, 94, 1372, 106, 118, 122, 24576, 134, 142, 146, 158, 17496, 166, 178, 194, 202, 206, 214, 218, 226, 5324, 18750, 254, 114688, 262, 274, 278, 298, 302, 314, 326, 334

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

A001787 and A024622 give record values and where they occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003415, A024622, A025473, A025474, A056798, A192084.

Subsequences: A001787, A027471, A051674, A079705, A100484.

s[n_] := If[PrimePowerQ[n], f = FactorInteger[n][[1]]; 2*f[[2]]*n^(2 - 1/f[[2]]), Nothing]; s[1] = 0; Array[s, 200] (* Amiram Eldar, Apr 06 2025 *)

a(n) = 2 * A025474(n) * A025473(n)^(2*A025474(n) - 1).

A192084(n) = A003415(a(n)).

cp's OEIS Frontend

A056798 Prime powers with even nonnegative exponents.

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