A269404 -id:A269404 - cp's OEIS frontend

A030516 Numbers with 7 divisors. 6th powers of primes.

64, 729, 15625, 117649, 1771561, 4826809, 24137569, 47045881, 148035889, 594823321, 887503681, 2565726409, 4750104241, 6321363049, 10779215329, 22164361129, 42180533641, 51520374361, 90458382169, 128100283921

Offset: 1

Jeff Burch

These are the numbers p^6 with p prime. - Jorge Coveiro, Apr 13 2004

The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

R. J. Mathar, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Subsequence of A046306.

Cf. A013664, A050997, A085966, A131993, A000005, A000040, A001248, A030514, A258603, A269404.

a030516 = (^ 6) . a000040
a030516_list = map (^ 6) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

[NthPrime(n)^6: n in [1..400]];  // Bruno Berselli, Jul 30 2011

Array[Prime[ # ]^6&,60] (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)

for(n=1,1e3,print1(prime(n)^6", "));  \\ Bruno Berselli, Jul 30 2011

[p**6 for p in prime_range(100)]
# Zerinvary Lajos, May 16 2007

a(n) = A000040(n)^(7-1) = A000040(n)^6. - Omar E. Pol, May 06 2008
A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(6) = 0.0170700868... (A085966). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(6)/zeta(12) = 675675/(691*Pi^6) (A269404).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(6) = 945/Pi^6 = 1/A013664. (End)

A113851 Numbers whose prime factors are raised to the sixth power.

Original entry on oeis.org

64, 729, 15625, 46656, 117649, 1000000, 1771561, 4826809, 7529536, 11390625, 24137569, 47045881, 85766121, 113379904, 148035889, 308915776, 594823321, 729000000, 887503681, 1291467969, 1544804416, 1838265625, 2565726409, 3010936384, 3518743761, 4750104241

Offset: 1

Cino Hilliard, Jan 25 2006

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Subset of A001014. Superset of A030516.
Nonunit terms of A329332 column 6 in ascending order.
Cf. A005117, A269404.

for n from 2 to 100 do if(numtheory[issqrfree](n))then printf("%d, ", n^6): fi: od: # Nathaniel Johnston, Jun 21 2011

Select[ Range@37^6, Union[Last /@ FactorInteger@# ] == {6} &] (* Robert G. Wilson v *)
Select[Range[2, 37], SquareFreeQ]^6 (* Amiram Eldar, Oct 13 2020 *)

from math import isqrt
from sympy import mobius
def A113851(n):
    def f(x): return int(n+1-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**6 # Chai Wah Wu, Feb 25 2025

a(n) = A005117(n+1)^6. - Nathaniel Johnston, Jun 21 2011

Sum_{n>=1} 1/a(n) = zeta(6)/zeta(12) - 1 = A269404 - 1. - Amiram Eldar, Oct 13 2020

More terms from Robert G. Wilson v, Jan 26 2006

cp's OEIS Frontend

A030516 Numbers with 7 divisors. 6th powers of primes.

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