cp's OEIS Frontend

A050997 Fifth powers of primes.

%I A050997 #87 Feb 16 2025 08:32:41
%S A050997 32,243,3125,16807,161051,371293,1419857,2476099,6436343,20511149,
%T A050997 28629151,69343957,115856201,147008443,229345007,418195493,714924299,
%U A050997 844596301,1350125107,1804229351,2073071593,3077056399,3939040643,5584059449,8587340257,10510100501
%N A050997 Fifth powers of primes.
%C A050997 Numbers k such that A062799(k) = 5.
%C A050997 Let r(n) = (a(n)+1)/(a(n)-1) if a(n) mod 4 = 3, (a(n)-1)/(a(n)+1) otherwise; then Product_{n>=1} r(n) = (31/33) * (244/242) * (3124/3126) * (16808/16806) * ... = 246016/259875. - _Dimitris Valianatos_, Mar 09 2020
%H A050997 T. D. Noe, <a href="/A050997/b050997.txt">Table of n, a(n) for n = 1..1000</a>
%H A050997 Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.
%H A050997 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">MathWorld: Prime Power</a>.
%H A050997 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A050997 A056595(a(n)) = 3. - _Reinhard Zumkeller_, Aug 15 2011
%F A050997 Sum_{n>=1} 1/a(n) = P(5) = 0.0357550174... (A085965). - _Amiram Eldar_, Jul 27 2020
%F A050997 From _Amiram Eldar_, Jan 23 2021: (Start)
%F A050997 Product_{n>=1} (1 + 1/a(n)) = zeta(5)/zeta(10) (A157291).
%F A050997 Product_{n>=1} (1 - 1/a(n)) = 1/zeta(5) = 1/A013663. (End)
%t A050997 Array[Prime[ # ]^5 &, 30] (* _Vladimir Joseph Stephan Orlovsky_, May 01 2008 *)
%o A050997 (PARI) vector(66,n,prime(n)^5)
%o A050997 (Magma) [p^5: p in PrimesUpTo(300)]; // _Vincenzo Librandi_, Mar 27 2014
%o A050997 (Haskell)
%o A050997 a050997 = (^ 5) . a000040
%o A050997 a050997_list = map (^ 5) a000040_list
%o A050997 -- _Reinhard Zumkeller_, Jun 03 2015
%Y A050997 Cf. A000040, A001248, A030078, A030514, A085965, A131992, A131993, A013663, A157291.
%Y A050997 Cf. A258602.
%K A050997 nonn,easy
%O A050997 1,1
%A A050997 _Eric W. Weisstein_