cp's OEIS Frontend

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

%I A030516 #88 Feb 04 2024 01:12:52
%S A030516 64,729,15625,117649,1771561,4826809,24137569,47045881,148035889,
%T A030516 594823321,887503681,2565726409,4750104241,6321363049,10779215329,
%U A030516 22164361129,42180533641,51520374361,90458382169,128100283921
%N A030516 Numbers with 7 divisors. 6th powers of primes.
%C A030516 These are the numbers p^6 with p prime. - _Jorge Coveiro_, Apr 13 2004
%C A030516 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
%H A030516 R. J. Mathar, <a href="/A030516/b030516.txt">Table of n, a(n) for n = 1..1000</a>
%H A030516 OEIS Wiki, <a href="https://oeis.org/wiki/Index_entries_for_number_of_divisors">Index entries for number of divisors</a>
%H A030516 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A030516 a(n) = A000040(n)^(7-1) = A000040(n)^6. - _Omar E. Pol_, May 06 2008
%F A030516 A056595(a(n)) = 3. - _Reinhard Zumkeller_, Aug 15 2011
%F A030516 Sum_{n>=1} 1/a(n) = P(6) = 0.0170700868... (A085966). - _Amiram Eldar_, Jul 27 2020
%F A030516 From _Amiram Eldar_, Jan 23 2021: (Start)
%F A030516 Product_{n>=1} (1 + 1/a(n)) = zeta(6)/zeta(12) = 675675/(691*Pi^6) (A269404).
%F A030516 Product_{n>=1} (1 - 1/a(n)) = 1/zeta(6) = 945/Pi^6 = 1/A013664. (End)
%t A030516 Array[Prime[ # ]^6&,60] (* _Vladimir Joseph Stephan Orlovsky_, Dec 14 2009 *)
%o A030516 (Sage)
%o A030516 [p**6 for p in prime_range(100)]
%o A030516 # _Zerinvary Lajos_, May 16 2007
%o A030516 (PARI) for(n=1,1e3,print1(prime(n)^6", "));  \\ _Bruno Berselli_, Jul 30 2011
%o A030516 (Magma) [NthPrime(n)^6: n in [1..400]];  // _Bruno Berselli_, Jul 30 2011
%o A030516 (Haskell)
%o A030516 a030516 = (^ 6) . a000040
%o A030516 a030516_list = map (^ 6) a000040_list
%o A030516 -- _Reinhard Zumkeller_, Jun 03 2015
%Y A030516 Subsequence of A046306.
%Y A030516 Cf. A013664, A050997, A085966, A131993, A000005, A000040, A001248, A030514, A258603, A269404.
%K A030516 nonn,easy
%O A030516 1,1
%A A030516 _Jeff Burch_