A160773 -id:A160773 - cp's OEIS frontend

A166241 Primes of the form a(n) = 3^n + 5^n + 7^n.

3, 83, 292299923, 684331371443, 191640836307771341507, 9388970456309004899603, 30363584636685952989516426809065192841572196335875312999038527044324069339846978313376123672504677550327603

Offset: 1

Zak Seidov, Oct 09 2009

nonn

The next term has 435 digits. - Harvey P. Dale, Apr 06 2015

Cf. A160773 (corresponding values of n).

[ a: n in [0..450]|IsPrime(a) where a is 3^n+5^n+7^n] // Vincenzo Librandi, Dec 08 2010

Select[Table[3^n+5^n+7^n,{n,0,200}],PrimeQ] (* Harvey P. Dale, Apr 06 2015 *)

A176613 Smallest prime p of three consecutive primes such that the sum of their n-th powers is prime, or 0 if such a prime does not exist.

Original entry on oeis.org

2, 5, 3, 23, 0, 11, 0, 5, 0, 23, 3, 137, 0, 5, 3, 89, 0, 71, 0, 17, 0, 23, 0, 23, 3, 131, 3, 419, 0, 31, 0, 859, 0, 31, 0, 127, 0, 11, 0, 359, 0, 31, 0, 347, 0, 509, 0, 137, 0, 193, 0, 769, 0, 23, 0, 17

Offset: 0

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 21 2010

nonn

Let p = prime(i), q = prime(i+1), r = prime(i+2).
(*) p^n + q^n + r^n has to be a prime.
When n is even and p > 3, then (*) is composite because primes greater than 3 are either of form 6k-1 or 6k+1 for some k. Hence, squares (or any even power) of such a prime has the form 6k+1. Adding three such even powers will produce a number of the form 6k+3, which is divisible by 3.
When n is even and p = 3, sequence A160773 gives the even n for which 3^n + 5^n + 7^n is prime.

			5 + 7 + 11 = 23 = prime(9); 3^2 + 5^2 + 7^2 = 83 = prime(23); 23^3 + 29^3 + 31^3 = 66347 = prime(6616).

Robert Israel, Table of n, a(n) for n = 0..500

Cf. A133530, A133531, A133532, A133533, A160773.

f:= proc(n) local p,q,r;
  if n::even then
    if isprime(3^n+5^n+7^n) then return 3
    else return 0
    fi
  fi;
  p:= 2: q:= 3: r:= 5:
  while not isprime(p^n + q^n + r^n) do
    p:= q; q:= r; r:= nextprime(r)
  od;
  p
end proc:
f(0):= 2:
map(f, [$0..100]);

a(0) term added by T. D. Noe, Nov 23 2010

A352393 Numbers k such that 3^k + 5^k + 7^k + 11^k + 13^k is prime.

Original entry on oeis.org

0, 2, 4, 6, 12, 14, 28, 60, 68, 2070, 7910, 10740

Offset: 1

Hemjyoti Nath, Jun 07 2022

Note that k must be even.

If it exists, a(13) > 31000. - Hugo Pfoertner, Jun 08 2022

			For k=2 we obtain f(2) = 3^2 + 5^2 + 7^2 + 11^2 + 13^2 = 373 which is a prime.

Cf. A354831, A160773.

Select[Range[0, 1000], PrimeQ[3^# + 5^# + 7^# + 11^# +13^#] &]

from sympy import isprime
from itertools import count, islice
def agen(): yield from (k for k in count(0) if isprime(3**k + 5**k + 7**k + 11**k + 13**k))
print(list(islice(agen(), 9))) # Michael S. Branicky, Jun 07 2022

a(11)-a(12) from Hugo Pfoertner, Jun 07 2022

cp's OEIS Frontend

A166241 Primes of the form a(n) = 3^n + 5^n + 7^n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

A176613 Smallest prime p of three consecutive primes such that the sum of their n-th powers is prime, or 0 if such a prime does not exist.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A352393 Numbers k such that 3^k + 5^k + 7^k + 11^k + 13^k is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Extensions