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

Author

Jeff Burch

Keywords

• nonn
• easy

Comments

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

Links

• 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

Crossrefs

Subsequence of A046306.

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

Programs

Haskell

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

Magma

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

Mathematica

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

PARI

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

Sage

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

Formula

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)