A007611 -id:A007611 - cp's OEIS frontend

A249945 a(n) = n! + 3^n.

2, 4, 11, 33, 105, 363, 1449, 7227, 46881, 382563, 3687849, 40093947, 479533041, 6228615123, 87183074169, 1307688716907, 20922832934721, 355687557236163, 6402374093148489, 121645101571093467, 2432902011663424401, 51090942182169793203, 1124000727808988739609

Offset: 0

Max Alweiss, Nov 08 2014

Cf. A000142, A000244, A007611.

[3^n+Factorial(n) : n in [0..25]]; // Wesley Ivan Hurt, Nov 14 2014

A249945:=n->3^n+n!: seq(A249945(n), n=0..25); # Wesley Ivan Hurt, Nov 14 2014

Table[3^n + n!, {n, 20}] (* Alonso del Arte, Nov 09 2014 *)

vector(10,n,3^n + n!) \\ Derek Orr, Nov 14 2014

a(n) = n! + 3^n.
a(n) = A000142(n) + A000244(n).
E.g.f.: 1/(1-x) + exp(3*x). - Alois P. Heinz, Aug 29 2022

A261714 Numbers n such that n! + 2^n + 1 is prime.

Original entry on oeis.org

0, 2, 4, 8, 72

Offset: 1

Altug Alkan, Aug 29 2015

Inspired by A007611.
First three prime numbers of the form n! + 2^n +1 with positive n value are 7, 41, 40577.
a(5) > 30000. - Giovanni Resta, Aug 30 2015

			For n=2, n! + 2^n + 1 = 2! + 2^2 + 1 = 7, which is prime.

Cf. A007611.

[n: n in [1..300] | IsPrime(Factorial(n)+2^n+1)]; // Vincenzo Librandi, Aug 30 2015

Select[Range@ 2000, PrimeQ[#! + 2^# + 1] &] (* Michael De Vlieger, Aug 29 2015 *)

for(n=1, 1e3, if(isprime(k=(n!+2^n+1)), print1(n", ")))

ABC2 $a! + 2^$a + 1
a: from 0 to 30000
Charles R Greathouse IV, Sep 08 2015

A263469 Numbers k such that k! + 2^k + 3 or k! + 2^k - 3 is prime.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 15, 17, 21, 42, 57, 99, 312, 372, 15030

Offset: 1

Altug Alkan, Oct 19 2015

Both k! + 2^k + 3 and k! + 2^k - 3 are prime for k = 3 or 4. Are there any others?

No more terms below 10^4. - Charles R Greathouse IV, Nov 17 2015

			For k = 0, k! + 2^k + 3 = 0! + 2^0 + 3 = 5, which is prime.
For k = 2, k! + 2^k - 3 = 2! + 2^2 - 3 = 3, which is prime.

Cf. A007611, A261714, A263482.

Select[Range[0, 10^3], Or[PrimeQ[#! + 2^# + 3], PrimeQ[#! + 2^# - 3]] &] (* Michael De Vlieger, Oct 20 2015 *)

for(n=0, 1e3, if(isprime(n!+2^n-3) || isprime(n!+2^n+3), print1(n", ")))

is(n)=my(N=n!+2^n); ispseudoprime(N-3) || ispseudoprime(N+3) \\ Charles R Greathouse IV, Nov 17 2015

a(14)-a(15) from Michael De Vlieger, Oct 20 2015

a(16) from Michael S. Branicky, Jul 25 2024

A263482 Numbers k such that k! + 2^k + 11 or k! + 2^k - 11 is prime.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 9, 15, 34, 41, 79, 99, 379, 2183

Offset: 1

Altug Alkan, Oct 19 2015

Is there some k such that k! + 2^k + 11 and k! + 2^k - 11 are prime?

a(16) > 20000. - Michael S. Branicky, Jul 25 2024

			For k = 0, k! + 2^k + 11 = 0! + 2^0 + 11 = 13, which is prime.
For k = 3, k! + 2^k - 11 = 3! + 2^3 - 11 = 3, which is prime.

Cf. A007611, A261714, A263469.

Select[Range[0, 400], Or[PrimeQ[#! + 2^# + 11], PrimeQ[#! + 2^# - 11]] &] (* Michael De Vlieger, Nov 17 2015 *)
Select[Range[0,500],AnyTrue[#!+2^#+{11,-11},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2019 *)

for(n=0, 1e3, if(isprime(n!+2^n-11) || isprime(n!+2^n+11), print1(n", ")))

is(n)=my(N=n!+2^n); ispseudoprime(N-11) || ispseudoprime(N+11) \\ Charles R Greathouse IV, Nov 17 2015

a(14) from Charles R Greathouse IV, Nov 17 2015

a(15) from Michael S. Branicky, Jun 17 2023

A269833 Numbers n such that 2^n + n! is the sum of 2 squares.

Original entry on oeis.org

0, 4, 6, 8, 16, 20, 21, 40, 45, 47, 52, 64, 67, 71, 72, 74, 88

Offset: 1

Altug Alkan, Mar 06 2016

Integers n such that the equation 2^n + n! = x^2 + y^2 where x and y are integers is solvable.
4, 8, 16 and 64 are powers of 2. What is the next power of 2 (if any) in this sequence?
103 <= a(18) <= 108. 108, 117, 144, 176, 254, 537 are terms. - Chai Wah Wu, Jul 22 2020

			6 is a term because 2^6 + 6! = 28^2.
8 is a term because 2^8 + 8! = 24^2 + 200^2.
21 is a term because 2^21 + 21! = 1222129664^2 + 7042537984^2.

Cf. A001481, A007611.

Select[Range[0, 64], SquaresR[2, 2^# + #!] > 0 &] (* Michael De Vlieger, Mar 07 2016 *)

isA001481(n) = #bnfisintnorm(bnfinit(z^2+1), n);
for(n=0, 1e2, if(isA001481(n!+2^n), print1(n, ", ")));

from math import factorial
from itertools import count, islice
from sympy import factorint
def A269833_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint((1<A269833_list = list(islice(A269833_gen(),9)) # Chai Wah Wu, Jun 27 2022

a(17) from Chai Wah Wu, Jul 22 2020

A308090 a(n) = gcd(2^n + n!, 3^n + n!, n+1).

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1

Offset: 1

Pedro Caceres, May 11 2019

nonn

From observation: For n > 3, if n+1 is prime, then a(n) = n+1.
This implies that (2^n + n!)= 0 mod (n+1) iff (n+1) is prime, and (3^n + n!)= 0 mod (n+1) iff (n+1) is prime.
Conjecture: Conversely, if gcd(2^n + n!, 3^n + n!, n+1) = n+1, then n+1 is prime.
Appears to be the same as A090585(n) except at n=2. - R. J. Mathar, Jul 22 2021

			a(4) = gcd(2^4 + 4!, 3^4 + 4!, 5) = gcd(40, 105, 5) = 5.
a(5) = gcd(2^5 + 5!, 3^5 + 5!, 6) = gcd(152, 363, 6) = 1.

Cf. A007611, A249945.

Table[GCD[2^n+n!,3^n+n!,n+1],{n,100}] (* Harvey P. Dale, Aug 27 2020 *)

a(n) = gcd([2^n + n!, 3^n + n!, n+1]); \\ Michel Marcus, May 12 2019

a(n) = gcd(A007611(n), A249945(n), n+1).

cp's OEIS Frontend

A249945 a(n) = n! + 3^n.

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A261714 Numbers n such that n! + 2^n + 1 is prime.

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A263469 Numbers k such that k! + 2^k + 3 or k! + 2^k - 3 is prime.

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A263482 Numbers k such that k! + 2^k + 11 or k! + 2^k - 11 is prime.

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A269833 Numbers n such that 2^n + n! is the sum of 2 squares.

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A308090 a(n) = gcd(2^n + n!, 3^n + n!, n+1).

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