A157291 -id:A157291 - cp's OEIS frontend

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

Eric W. Weisstein

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

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

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

Cf. A258602.

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

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

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

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

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)

A113850 Numbers whose prime factors are raised to the fifth power.

Original entry on oeis.org

32, 243, 3125, 7776, 16807, 100000, 161051, 371293, 537824, 759375, 1419857, 2476099, 4084101, 5153632, 6436343, 11881376, 20511149, 24300000, 28629151, 39135393, 45435424, 52521875, 69343957, 79235168, 90224199, 115856201

Offset: 1

Cino Hilliard, Jan 25 2006

			7776 = 32*243 = 2^5*3^5 so the prime factors, 2 and 3, are raised to the fifth power.

Proper subset of A000584.
Nonunit terms of A329332 column 5 in ascending order.
Cf. A005117, A157291.

Select[ Range@41^5, Union[Last /@ FactorInteger@# ] == {5} &] (* Robert G. Wilson v *)
Rest[Select[Range[100], SquareFreeQ]^5] (* Vaclav Kotesovec, May 22 2020 *)

allpwrfact(n,p) = \All prime factors are raised to the power p { local(x,j,ln,y,flag); for(x=4,n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1,ln, if(y[2][j]==p,flag++); ); if(flag==ln,print1(x",")); ) }

from math import isqrt
from sympy import mobius
def A113850(n):
    def f(x): return int(n+x+1-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**5 # Chai Wah Wu, Sep 13 2024

Sum_{k>=1} 1/a(k) = zeta(5)/zeta(10) - 1 = A157291 - 1. - Amiram Eldar, May 22 2020

a(n) = A005117(n+1)^5. - Chai Wah Wu, Sep 13 2024

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

Offset corrected by Chai Wah Wu, Sep 13 2024

A347330 Decimal expansion of zeta(10) / zeta(5).

Original entry on oeis.org

9, 6, 5, 3, 4, 6, 4, 9, 6, 0, 9, 1, 6, 3, 6, 0, 3, 6, 2, 1, 3, 7, 7, 2, 9, 6, 4, 2, 4, 3, 2, 2, 1, 2, 2, 4, 7, 4, 0, 5, 0, 1, 6, 0, 5, 3, 1, 8, 7, 3, 0, 1, 8, 0, 1, 5, 7, 5, 6, 4, 6, 4, 7, 2, 6, 8, 8, 1, 8, 6, 5, 2, 4, 4, 3, 9, 9, 0, 6, 4, 8, 0, 5, 4, 8, 3, 8

Offset: 0

Sean A. Irvine, Aug 26 2021

			0.96534649609163603621377296424322122474050...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A008836, A182448, A347328, A347329, A347331.

Cf. A013663, A013668.

RealDigits[Zeta[10] / Zeta[5], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals Sum_{k>=1} A008836(k) / k^5.
Equals Product_{p prime} 1/(1+p^(-5)). [corrected by Amiram Eldar, Jun 06 2023]
Equals 1/A157291. - R. J. Mathar, Jul 20 2025

A269404 Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).

Original entry on oeis.org

1, 0, 1, 7, 0, 9, 2, 7, 6, 9, 1, 3, 0, 4, 9, 9, 2, 7, 6, 6, 4, 3, 2, 7, 2, 1, 3, 3, 0, 9, 7, 9, 0, 9, 9, 2, 0, 4, 9, 2, 2, 1, 9, 0, 7, 9, 4, 9, 4, 1, 0, 1, 1, 3, 4, 6, 6, 4, 6, 5, 1, 7, 9, 3, 8, 1, 8, 9, 3, 5, 3, 3, 5, 8, 3, 4, 2, 2, 7, 9, 4, 3, 1, 8, 1, 5, 1, 5, 9, 6, 4, 7, 8, 5, 0, 6, 6, 8, 9, 7, 8, 4, 5, 4, 6, 5, 1, 0, 6, 4, 0, 2, 6, 1, 3, 3, 6, 9, 3, 0

Offset: 1

Ilya Gutkovskiy, Feb 25 2016

More generally, Product_{k >= 1} (1 + 1/prime(k)^m) = zeta(m)/zeta(2*m), where zeta(m) is the Riemann zeta function.

			1.0170927691304992766432721330979099204922190794941...

Eric Weisstein's World of Mathematics, Prime Products.

Cf. A005117, A082020, A113851, A157289, A157290, A157291.

RealDigits[Zeta[6]/Zeta[12], 10, 120][[1]]
RealDigits[675675/(691 Pi^6), 10, 120][[1]]

zeta(6)/zeta(12) \\ Amiram Eldar, Jun 11 2023

Equals zeta(6)/zeta(12).
Equals 675675/(691*Pi^6).
Equals Sum_{k>=1} 1/A005117(k)^6 = 1 + Sum_{k>=1} 1/A113851(k). - Amiram Eldar, Jun 27 2020

cp's OEIS Frontend

A050997 Fifth powers of primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

A113850 Numbers whose prime factors are raised to the fifth power.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A347330 Decimal expansion of zeta(10) / zeta(5).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A269404 Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula