A268067 -id:A268067 - cp's OEIS frontend

A076515 Numbers k such that 1 + 3^k + 5^k is prime.

0, 12, 36, 48, 72, 120, 605376

Offset: 1

Zak Seidov, Oct 17 2002

Next term, if it exists, is greater than 35000. - Vaclav Kotesovec, Jan 26 2016
No more terms up to 75000 (previous range rechecked). All terms are multiples of 12: if k > 0 is not a multiple of 12, 1 + 3^k + 5^k is divisible by 3, 5 or 7. - Rick L. Shepherd, Aug 06 2017
Next term, if it exists, is greater than 10^6. - Jon Grantham, Jul 29 2023

Cf. A008594, A074507, A268064, A268065, A268067.

[n: n in [0..1000]|IsPrime(3^n+5^n+1)] // Vincenzo Librandi, Jan 22 2011

A076515:=n->`if`(isprime(1+3^n+5^n), n, NULL): seq(A076515(n), n=0..200); # Wesley Ivan Hurt, Aug 06 2017

Do[ If[ PrimeQ[1 + 3^n + 5^n], Print[n]], {n, 0, 3500, 2}]
Select[Range[0,5000],PrimeQ[1+3^#+5^#]&] (* Harvey P. Dale, Mar 09 2012 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(1 + 3^n + 5^n), print1(n, ", "))); \\ Altug Alkan, Jan 25 2016

a(7) from Jon Grantham, Jul 29 2023

A268064 Integers k such that (2^k + 1) + (3^k + 1) + (5^k + 1) is prime.

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 30, 34, 35, 46, 55, 62, 83, 230, 1675, 2551, 3934, 25101, 28703, 46295, 54363, 72846

Offset: 1

Altug Alkan, Jan 25 2016

Inspired by A268067.
Integers k such that A000051(k) + A034472(k) + A034474(k) is a prime number.
Corresponding primes are 13, 41, 163, 16421, 9825701, 30531959803, 931322780507684352101, 582076625811872951801381, 2910383095704949820066203, ...
a(18) > 10000. - Tyler NeSmith, May 07 2021

			2 is a term because (2^2 + 1) + (3^2 + 1) + (5^2 + 1) = 41 is prime.

Cf. A000051, A034472, A034474, A268067.

Select[Range[0, 2000], PrimeQ[(2^#+1) + (3^#+1) + (5^#+1)] &] (* Vaclav Kotesovec, Jan 26 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(3 + 2^n + 3^n + 5^n), print1(n, ", ")));

a(18)-a(19) from Michael S. Branicky, Apr 12 2023

a(20)-a(22) from Michael S. Branicky, Sep 18 2024

A268065 Integers k such that 2^k + 3^k + 5^k + k is prime.

Original entry on oeis.org

0, 1, 3, 7, 19, 33, 1363, 6663

Offset: 1

Emre APARI, Jan 25 2016

a(8), if it exists, is greater than 30000. - Vaclav Kotesovec, Jan 26 2016

a(8), if it exists, is greater than 100000. - Michael S. Branicky, Dec 01 2024

			k=3: 2^3+3^3+5^3+3 = 163.

Cf. A076515, A268067.

Select[Range[0, 2000], PrimeQ[2^# + 3^# + 5^# + #] &] (* Vaclav Kotesovec, Jan 25 2016 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(2^n + 3^n + 5^n + n), print1(n, ", "))); \\ Altug Alkan, Jan 25 2016

a(7) from Vaclav Kotesovec, Jan 25 2016

A261722 Values of m such that 2^m + 3^m + 5^m + 7^m + 11^m + 13^m is a prime number.

Original entry on oeis.org

1, 7, 25, 91

Offset: 1

Altug Alkan, Aug 29 2015

2, 3, 5, 7, 11, 13 are first six consecutive prime numbers.
From Bruno Berselli, Sep 04 2015: (Start)
All terms are odd. In fact, assuming m even and b(k) = 4^k + 9^k + 25^k + 49^k + 121^k + 169^k, for
. k == 0, 2, 4 (mod 6), b(k) is divisible by 5;
. k == 1, 5 (mod 6), b(k) is divisible by 377 = 13*29;
. k == 3 (mod 6), b(k) is divisible by 29. (End)
From Jon E. Schoenfield, Mar 02 2018: (Start)
For n odd:
Let t(n) = 2^n + 3^n + 5^n + 7^n + 11^n + 13^n; then t(n) is divisible by prime p for certain pairs (p, n mod (p-1)):
.
   p  n mod (p-1) such that p|t(n)
  ==  ============================
   2               -
   3               -
   5               -
   7               -
  11               9
  13               -
  17               5
  19               9
  23              11
  29               3
  31              15
  37            21, 29
  41             1, 19
  43          11, 33, 37
  47              23
  53               -
  59            29, 55
  ...
The smallest prime p that divides t(n) at more than three values of n mod (p-1) is 313: 313|t(n) when n mod 312 is any of the four values {39, 117, 195, 273}, i.e., when n mod (312/4 = 78) = 39.
The smallest prime p that divides t(n) at more than four values of n mod (p-1) is 3041: 3041|t(n) when n mod 3040 is any of the 16 values {95, 285, 475, 665, 855, 1045, 1235, 1425, 1615, 1805, 1995, 2185, 2375, 2565, 2755, 2945}, i.e., when n mod (3040/16 = 190) = 95. (End)
No other terms than the four terms cited less than 25000. - Robert G. Wilson v, Mar 07 2018
No other terms than the four terms cited less than 100000. - Michael S. Branicky, Sep 28 2024

			1 is a term because 2^1 + 3^1 + 5^1 + 7^1 + 11^1 + 13^1 = 41 and 41 is a prime number.

Cf. A000040, A076515, A268064, A268067.

[n: n in [0..1000] | IsPrime(a) where a is 2^n+3^n+5^n+ 7^n+11^n+13^n]; // Vincenzo Librandi, Aug 30 2015

Select[Table[{n, Sum[Prime[k]^n, {k, 6}]}, {n, 1000}], PrimeQ[#[[2]]]&] [[All, 1]] (* Michael De Vlieger, Aug 29 2015 *)

for(n=1, 1e3, if(isprime(13^n+11^n+7^n+5^n+3^n+2^n), print1(n", ")))

Mathematica scripts updated by Jean-François Alcover, Sep 04 2015

A270817 Integers k such that (2^k - 1) + (3^k - 1) + (5^k - 1) is prime.

Original entry on oeis.org

1, 3, 4, 9, 11, 69, 117, 449, 675, 1119, 1959, 2687, 2859, 8001, 8175, 24269, 110247

Offset: 1

Altug Alkan, Mar 23 2016

Inspired by A268067.

Corresponding primes are 7, 157, 719, 1973317, 49007317, ...

			4 is a term because (2^4 - 1) + (3^4 - 1) + (5^4 - 1) = 719 is a prime number.

Cf. A268064, A268067.

Select[Range@ 3000, PrimeQ[(2^# - 1) + (3^# - 1) + (5^# - 1)] &] (* Michael De Vlieger, Mar 23 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(-3 + 2^n + 3^n + 5^n), print1(n, ", ")));

from sympy import isprime
def afind(limit, startk=1):
    pow2, pow3, pow5 = 2**startk, 3**startk, 5**startk
    for k in range(startk, limit+1):
        if isprime(pow2 + pow3 + pow5 - 3): print(k, end=", ")
        pow2 *= 2; pow3 *= 3; pow5 *= 5
afind(1200) # Michael S. Branicky, Sep 08 2021

a(14)-a(15) from Michael S. Branicky, Sep 08 2021
a(16) from Michael S. Branicky, Apr 13 2023
a(17) from Michael S. Branicky, Nov 27 2024

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A076515 Numbers k such that 1 + 3^k + 5^k is prime.

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A268064 Integers k such that (2^k + 1) + (3^k + 1) + (5^k + 1) is prime.

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A268065 Integers k such that 2^k + 3^k + 5^k + k is prime.

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A261722 Values of m such that 2^m + 3^m + 5^m + 7^m + 11^m + 13^m is a prime number.

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Magma

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A270817 Integers k such that (2^k - 1) + (3^k - 1) + (5^k - 1) is prime.

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