A057469 -id:A057469 - cp's OEIS frontend

A125956 Numbers k such that (2^k + 9^k)/11 is prime.

3, 7, 127, 283, 883, 1523, 4001

Offset: 1

Alexander Adamchuk, Feb 06 2007

All terms are primes. Note that first 3 terms {3, 7, 127} are primes of the form 2^q - 1, where q = {2, 3, 7} is prime too. Corresponding primes of the form (2^k + 9^k)/11 are {67, 434827, ...}.

a(8) > 10^5. - Robert Price, Dec 23 2012

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime.
Cf. A057469 = numbers n such that (2^n + 3^n)/5 is prime.
Cf. A082387 = numbers n such that (2^n + 5^n)/7 is prime.

Do[p=Prime[n];f=(2^p+9^p)/11; If[PrimeQ[f], Print[{p, f}]], {n, 1, 100}]

is(n)=ispseudoprime((2^n+9^n)/11) \\ Charles R Greathouse IV, Feb 20 2017

2 more terms from Rick L. Shepherd, Feb 14 2007

a(7) from Robert Price, Dec 23 2012

A125957 Numbers n such that (2^n + 11^n)/13 is prime.

Original entry on oeis.org

3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781

Offset: 1

Alexander Adamchuk, Feb 06 2007

All terms are primes. Corresponding primes of the form (2^n + 11^n)/13 are {103, 12391, 38880540653801911, ...}.

a(11) > 10^5. - Robert Price, Feb 26 2013

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A057469 = numbers n such that (2^n + 3^n)/5 is prime. Cf. A082387 = numbers n such that (2^n + 5^n)/7 is prime.

Do[p=Prime[n];f=(2^p+11^p)/13; If[PrimeQ[f], Print[{p, f}]], {n, 1, 100}]

is(n)=ispseudoprime((2^n+11^n)/13) \\ Charles R Greathouse IV, Feb 20 2017

2 more terms from Ryan Propper, Feb 09 2008

a(9)-a(10) from Robert Price, Feb 26 2013

A227170 Numbers n such that (16^n + 15^n)/31 is prime.

Original entry on oeis.org

3, 5, 13, 1439, 1669, 37691

Offset: 1

Jean-Louis Charton, Jul 03 2013

All terms are prime.

a(7) > 10^5. - Robert Price, Aug 26 2013

Cf. A000978, A057469, A128066, A128335, A128336, A187805, A181141, A187819, A217095.

is(n)=ispseudoprime((16^n+15^n)/31) \\ Charles R Greathouse IV, May 22 2017

A125958 Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 7, 3, 5, 5, 11, 3, 19, 11, 3, 229, 47, 5, 257, 3, 19, 31, 17, 11, 13, 3, 3, 5, 5, 59, 23, 3, 3, 7, 79, 3, 3373, 3, 3, 7, 13, 7, 7, 3527, 593, 19, 3, 3, 13, 13, 11, 19, 41, 3, 7, 109, 3, 227, 13, 5, 5, 3, 239, 5, 3251, 3, 1237, 3, 7, 31, 3, 7

Offset: 1

Alexander Adamchuk, Feb 06 2007

All terms are odd primes.
a(38),...,a(43) = {3,3,7,13,7,7}.
a(46),...,a(64) = {19,3,3,13,13,11,19,41,3,7,109,11,227,13,5,5,3,239,5}.
a(66) = 3. a(68),...,a(72) = {3,7,31,3,7}.
a(74),...,a(92) = {3,5,19,17,3,83,3,3,19,19,11,11,61,3,7,7,3,41,29}.
a(94) = 5. a(97),a(98) = {19,7}. a(100) = 31.
a(n) is currently unknown for n = {37,44,45,65,67,73,93,95,96,99,...}.
From Kevin P. Thompson, May 18 2022: (Start)
All known terms from n=1..100 correspond to proven primes.
a(96) > 10250.
a(99) > 10250. (End)
Presuming every prime is seen at least once, one can specifically seek those with fixed k. Doing this, a(174) = 37, a(368) = 43 for example. - Bill McEachen, Aug 26 2024

			For n=4, the expression (2^k + (2n-1)^k)/(2n+1) takes on values 1, 53/9, 39, 2417/9, and 1871 for k=1..5. Since 1871 is the first prime number to occur, a(4) = 5.

Kevin P. Thompson, Table of n, a(n) for n = 1..95

Cf. A000978 ((2^n + 1)/3 is prime), A057469 ((2^n + 3^n)/5 is prime).
Cf. A082387 ((2^n + 5^n)/7 is prime), A125955 ((2^n + 7^n)/9 is prime).
Cf. A125956 ((2^n + 9^n)/11 is prime), A125955 ((2^n + 11^n)/13 is prime).

Do[k = 1; While[ !PrimeQ[(2^k + (2n-1)^k)/(2n+1)], k++ ]; Print[k], {n, 100}] (* Ryan Propper, Mar 29 2007 *)

More terms from Ryan Propper, Mar 29 2007

a(65)-a(72) from Kevin P. Thompson, May 18 2022

A247244 Smallest prime p such that (n^p + (n+1)^p)/(2n+1) is prime, or -1 if no such p exists.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 7, 3, 7, 53, 47, 3, 7, 3, 3, 41, 3, 5, 11, 3, 3, 11, 11, 3, 5, 103, 3, 37, 17, 7, 13, 37, 3, 269, 17, 5, 17, 3, 5, 139, 3, 11, 78697, 5, 17, 3671, 13, 491, 5, 3, 31, 43, 7, 3, 7, 2633, 3, 7, 3, 5, 349, 3, 41, 31, 5, 3, 7, 127, 3, 19, 3, 11, 19, 101, 3, 5, 3, 3

Offset: 1

Eric Chen, Nov 28 2014

nonn

All terms are odd primes.
a(79) > 10000, if it exists.
a(80)..a(93) = {3, 7, 13, 7, 19, 31, 13, 163, 797, 3, 3, 11, 13, 5}, a(95)..a(112) = {5, 2657, 19, 787, 3, 17, 3, 7, 11, 1009, 3, 61, 53, 2371, 5, 3, 3, 11}, a(114)..a(126) = {103, 461, 7, 3, 13, 3, 7, 5, 31, 41, 23, 41, 587}, a(128)..a(132) = {7, 13, 37, 3, 23}, a(n) is currently unknown for n = {79, 94, 113, 127, 133, ...} (see the status file under Links).

			a(10) = 53 because (10^p + 11^p)/21 is composite for all p < 53 and prime for p = 53.

Aurelien Gibier, Table of n, a(n) for n = 1..78 (first 42 terms from Robert G. Wilson v)
Robert G. Wilson v, Table of n, a(n) for n = 1..1000 with status of unknown terms [updated/edited by Jon E. Schoenfield]

Cf. A058013, A125713, A000978, A057469, A128066, A128335, A128336, A187805, A181141, A187819, A217095, A185239, A213216, A225097, A224984, A221637, A227170, A228573, A227171, A225818, A227172, A227173, A227174.

lmt = 4200; f[n_] := Block[{p = 2}, While[p < lmt && !PrimeQ[((n + 1)^p + n^p)/(2n + 1)], p = NextPrime@ p]; If[p > lmt, 0, p]]; Do[Print[{n, f[n] // Timing}], {n, 1000}] (* Robert G. Wilson v, Dec 01 2014 *)

a(n)=forprime(p=3, , if(ispseudoprime((n^p+(n+1)^p)/(2*n+1)), return(p)))

a(n) = 3 if and only if n^2 + n + 1 is a prime (A002384).

a(43) from Aurelien Gibier, Nov 27 2023

A181628 Numbers k such that (2^k + 3^k)/13 is prime.

Original entry on oeis.org

6, 10, 14, 22, 34, 38, 82, 106, 218, 334, 4414, 7246, 10118, 10942, 15898, 42422, 65986

Offset: 1

Michel Lagneau, Nov 18 2010

All terms are of the form 2p, p prime.
The prime (2^4414 + 3^4414)/13 = 79300327387 ...611266 985181 has 2105 decimal digits.
a(18) > 10^5. - Michael S. Branicky, Aug 17 2024

			10 is in the sequence because (2^10+ 3^10)/13 = 60073/13 = 4621 is prime.

Cf. A057469.

with(numtheory):for n from 1 to 4500 do: x:= (2^n + 3^n)/13:if floor(x)=x and
  type(x,prime)=true then printf(`%d, `, n):else fi:od:
# alternative
Res:= NULL:
p:= 2:
while p < 6000 do
p:= nextprime(p);
if isprime((2^(2*p)+3^(2*p))/13) then Res:= Res, 2*p fi;
od:
Res; # Robert Israel, Apr 26 2017

is(n)=n%2==0 && isprime(n/2) && ispseudoprime((2^n+3^n)/13) \\ Charles R Greathouse IV, Jun 06 2017

from sympy import isprime
def afind(limit, startk=1):
    k = startk
    pow2 = 2**k
    pow3 = 3**k
    for k in range(startk, limit+1):
        q, r = divmod(pow2+pow3, 13)
        if r == 0 and isprime(q):
            print(k, end=", ")
        pow2 *= 2
        pow3 *= 3
afind(1000) # Michael S. Branicky, Dec 28 2021

a(12) from D. S. McNeil, Nov 18 2010
a(13) and a(14) from Robert Israel, Apr 26 2017
a(15) from Michael S. Branicky, Dec 28 2021
a(16) from Michael S. Branicky, Apr 26 2023
a(17) from Michael S. Branicky, Aug 17 2024

A227171 Numbers n such that (18^n + 17^n)/35 is prime.

Original entry on oeis.org

3, 47, 53, 2411, 4057, 7963, 10273, 15737, 53299

Offset: 1

Jean-Louis Charton, Jul 03 2013

All terms are prime.

Cf. A000978, A057469, A128066, A128335, A128336, A187805, A181141, A187819, A217095.

is(n)=ispseudoprime((18^n+17^n)/35) \\ Charles R Greathouse IV, Jun 13 2017

a(7), a(8) from Richard Fischer, Aug 18 2013

a(9) from Robert Price, Aug 25 2013

A227979 Integers not of the form (a^k+b^k)/(a+b) for any positive integer values of a, b, k with b > a.

Original entry on oeis.org

2, 4, 6, 8, 9, 14, 16, 18, 22, 23, 24, 32, 33, 36, 38, 42, 44, 46, 47, 54, 56, 59, 62, 64, 66, 69, 71, 72, 77, 81, 83, 86, 88, 92, 94, 96, 98, 99, 107, 114, 118, 121, 126, 128, 131, 132, 134, 138, 141, 142, 144, 152, 154, 158, 161, 162, 166, 167, 168, 177

Offset: 1

Robert Price, Sep 30 2013

nonn

This form, (a^k+b^k)/(a+b), is a generalization of the Fermat numbers.
Not all integers are in this set.
See A229791 for the complement of this sequence.

Robert Price, Table of n, a(n) for n = 1..66
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

A few of the sequences using this form that identify primes are A000978, A007658, A057469, A128066, A057171, A082387, A122853, A128335.

limit=200; lst = {}; Do[p = (a^k + b^k)/(a + b); If[p <= limit && IntegerQ[p], AppendTo[lst, p]], {k, Log[2,3*limit+1]}, {b, 2, limit*2}, {a, b-1}]; Complement[Range[limit], Union[lst]]

A229791 Integers generated by (a^k+b^k)/(a+b) for all possible positive integer values of a,b,k with b>a.

Original entry on oeis.org

1, 3, 5, 7, 10, 11, 12, 13, 15, 17, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 34, 35, 37, 39, 40, 41, 43, 45, 48, 49, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 65, 67, 68, 70, 73, 74, 75, 76, 78, 79, 80, 82, 84, 85, 87, 89, 90, 91, 93, 95, 97, 100, 101, 102, 103

Offset: 1

Robert Price, Sep 29 2013

nonn

This form, (a^k+b^k)/(a+b), is a generalization of the Fermat numbers.
Not all integers are in this set.
See A227979 for the complement of this sequence.

Robert Price, Table of n, a(n) for n = 1..134
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

A few of the sequences using this form that identify primes are A000978, A007658, A057469, A128066, A057171, A082387, A122853, A128335.

limit=105; lst = {}; Do[p = (a^k + b^k)/(a + b); If[p <= limit && IntegerQ[p], AppendTo[lst, p]], {k, Log[2,3*limit+1]}, {b, 2, limit*2}, {a, b-1}]; Union[lst]

A125955 Numbers k such that (2^k + 7^k)/9 is prime.

Original entry on oeis.org

5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011

Offset: 1

Alexander Adamchuk, Feb 06 2007

All terms are primes. Corresponding primes of the form (2^k + 7^k)/9 are {1871, 3040971926676589439, 5469081705798319217773539465593130845206220817280793349743311, ...}.

a(12) > 10^5. - Robert Price, Aug 28 2012

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A057469 = numbers n such that (2^n + 3^n)/5 is prime. Cf. A082387 = numbers n such that (2^n + 5^n)/7 is prime.

Do[p=Prime[n];f=(2^p+7^p)/9; If[PrimeQ[f], Print[{p, f}]], {n, 1, 1000}]

is(n)=ispseudoprime((2^n+7^n)/9) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Ryan Propper, Mar 23 2007

cp's OEIS Frontend

A125956 Numbers k such that (2^k + 9^k)/11 is prime.

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A125957 Numbers n such that (2^n + 11^n)/13 is prime.

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A227170 Numbers n such that (16^n + 15^n)/31 is prime.

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A125958 Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.

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A247244 Smallest prime p such that (n^p + (n+1)^p)/(2n+1) is prime, or -1 if no such p exists.

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A181628 Numbers k such that (2^k + 3^k)/13 is prime.

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A227171 Numbers n such that (18^n + 17^n)/35 is prime.

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A227979 Integers not of the form (a^k+b^k)/(a+b) for any positive integer values of a, b, k with b > a.

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