A050997 Fifth powers of primes.

32, 243, 3125, 16807, 161051, 371293, 1419857, 2476099, 6436343, 20511149, 28629151, 69343957, 115856201, 147008443, 229345007, 418195493, 714924299, 844596301, 1350125107, 1804229351, 2073071593, 3077056399, 3939040643, 5584059449, 8587340257, 10510100501

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

Numbers k such that A062799(k) = 5.

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
• Eric Weisstein's World of Mathematics, MathWorld: Prime Power.
• Index to sequences related to prime signature

Crossrefs

Cf. A000040, A001248, A030078, A030514, A085965, A131992, A131993, A013663, A157291.

Cf. A258602.

Programs

Haskell

a050997 = (^ 5) . a000040
a050997_list = map (^ 5) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

Magma

[p^5: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Mathematica

Array[Prime[ # ]^5 &, 30] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

PARI

vector(66,n,prime(n)^5)

Formula

A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011

Sum_{n>=1} 1/a(n) = P(5) = 0.0357550174... (A085965). - Amiram Eldar, Jul 27 2020

From Amiram Eldar, Jan 23 2021: (Start)

Product_{n>=1} (1 + 1/a(n)) = zeta(5)/zeta(10) (A157291).

Product_{n>=1} (1 - 1/a(n)) = 1/zeta(5) = 1/A013663. (End)