A082101 -id:A082101 - cp's OEIS frontend

A094494 Primes p such that 2^j+p^j are primes for j=0,2,4,8.

6203, 16067, 72367, 105653, 179743, 323903, 1005467, 1040113, 1276243, 1331527, 1582447, 1838297, 1894873, 2202433, 2314603, 2366993, 2369033, 2416943, 2533627, 2698697, 2804437, 2806613, 2823277, 2826337, 2851867, 2888693, 3911783, 4217617, 4432837, 4475473

Offset: 1

Labos Elemer, Jun 01 2004

nonn

Primes of 2^j+p^j form are a generalization of Fermat-primes. 1^j is replaced by p^j. This is strongly supported by the observation that corresponding j-exponents are apparently powers of 2 like for the 5 known Fermat primes. See A094473-A094491.

			Conditions mean 2,p^2+4,16+p^4,256+p^8 are all primes.

Robert Israel, Table of n, a(n) for n = 1..400

Cf. A082101, A094473-A094492.

p:= 2: count:= 0: Res:= NULL:
while count < 30 do
  p:= nextprime(p);
  if isprime(4+p^2) and isprime(16+p^4) and isprime(256+p^8) then
    count:= count+1;
    Res:= Res, p;
  fi
od:
Res; # Robert Israel, Jul 17 2018

{ta=Table[0, {100}], u=1}; Do[s0=2;s2=4+Prime[j]^2;s2=16+Prime[j]^4;s8=256+Prime[j]^8 If[PrimeQ[s0]&&PrimeQ[s2]&&PrimeQ[s4]&&PrimeQ[s8], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[210000]],AllTrue[Table[2^j+#^j,{j,{0,2,4,8}}], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 13 2017 *)

A082102 Primes of the form 2^(k-1) + 3^k.

Original entry on oeis.org

11, 31, 89, 761, 2251, 6689, 59561, 14365291, 10461401779, 282437925089, 150094669656737489, 239299329512092506300739, 2153693964201457673153371, 107752636643058216783050887908587014548761

Offset: 1

Labos Elemer, Apr 14 2003

nonn

			89 = 2^3 + 3^4.

Harvey P. Dale, Table of n, a(n) for n = 1..30

Cf. A082101, A082103.

Do[s=2^(w-1)+3^w; If[PrimeQ[s], Print[s]], {w, 1, 2000}]
Select[Table[2^(n-1)+3^n,{n,100}],PrimeQ] (* Harvey P. Dale, Feb 09 2016 *)

lista(kmax) = {my(p); for(k = 1, kmax, p = 2^(k-1) + 3^k; if(isprime(p), print1(p, ", ")));} \\ Amiram Eldar, Aug 16 2024

a(n) = 2^(A082103(n)-1) + 3^A082103(n). - Amiram Eldar, Aug 16 2024

A093988 Numbers k such that 2^k + 3*k is prime.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 35, 45, 47, 57, 87, 183, 325, 367, 447, 809, 1157, 2789, 5775, 14829, 20687, 46463, 62491, 92147, 128745

Offset: 1

Robert G. Wilson v, May 25 2004

a(22) > 31410. - Jinyuan Wang, Feb 03 2020

Cf. A082101, A086653, A094963.

A093988:=n->`if`(isprime(2^n+3*n), n, NULL): seq(A093988(n), n=1..10^3); # Wesley Ivan Hurt, Jan 21 2017

Do[ If[ PrimeQ[2^n + 3n], Print[n]], {n, 1, 5000, 2}]

isok(n) = isprime(2^n + 3*n); \\ Michel Marcus, Jan 21 2017

ABC2 2^$a+3*$a
a: from 1 to 1000 // Jinyuan Wang, Feb 03 2020

a(19)-a(21) from Ryan Propper, Jul 05 2005
a(22)-a(23) from Michael S. Branicky, May 19 2023
a(24)-a(25) from Michael S. Branicky, Jul 24 2024

A094482 Primes of form 2^j + 137^j.

Original entry on oeis.org

2, 139, 18773, 124097929967680577

Offset: 1

Labos Elemer, Jun 01 2004

nonn

The number j must be zero or a power of 2. Checked j being powers of two through 2^20. Thus a(5) > 10^4400000. Primes of this magnitude are rare (about 1 in 10.3 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 29 2013

			j=0: p=1+1=2; j=1: p=2+59=61; j=2: p=4+18769=18773; j=8: p=256+37^8=124097929967680577; the j exponents are powers of 2.

Cf. A082101, A094473-A094480, A045637.

A161470 Primes of the form 3^k+2^k+k^3-k^2.

Original entry on oeis.org

5, 17, 53, 2609, 1604543, 7625731721669, 67585198634826967968486182915129003

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 10 2009

nonn

The associated k are 1, 2, 3, 7, 13, 27, 73, 994, 1129, ... - R. J. Mathar, Jun 12 2009

The next term has 475 digits. - Harvey P. Dale, Dec 12 2010

Cf. A058765, A082101, A161469.

[ a: n in [1..450] | IsPrime(a) where a is 3^n+2^n+n^3-n^2] // Vincenzo Librandi, Nov 30 2010

A161470 = {}; Do[If[PrimeQ[p = (3^n + 2^n) + (n^3 - n^2)], AppendTo[A161470, p]], {n, 6!}]; A161470 (* Orlovsky *)
(* Alternate: *) Select[Table[3^k + 2^k + k^3 - k^2, {k, 2000}], PrimeQ] (* Harvey P. Dale, Dec 12 2010 *)

Definition simplified by R. J. Mathar, Jun 12 2009

A165259 Sum of odd powers of 4 and 9 divided by 13.

Original entry on oeis.org

1, 61, 4621, 369181, 29821741, 2414250301, 195533302861, 15837861987421, 1282861452271981, 103911691734684541, 8416845656119913101, 681764476155480405661, 55222922216750191970221

Offset: 0

Jaume Oliver Lafont, Sep 11 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..520
Index entries for linear recurrences with constant coefficients, signature (97,-1296).

Cf. A007689, A082101, A096951.

[(4^(2*n+1)+9^(2*n+1))/13: n in [0..30]]; // G. C. Greubel, Mar 11 2023

Table[(4^n+9^n)/13,{n,1,31,2}] (* or *) LinearRecurrence[{97,-1296},{1,61},20] (* Harvey P. Dale, Jun 23 2013 *)

PARI
```
a(n)=(4^(2*n+1)+9^(2*n+1))/13
```

[(4^(2*n+1)+9^(2*n+1))/13 for n in range(31)] # G. C. Greubel, Mar 11 2023

G.f.: (1-36*x)/((1-16*x)*(1-81*x))
a(n) = 97*a(n-1) - 1296*a(n-2). - Harvey P. Dale, Jun 23 2013
E.g.f.: (1/13)*(4*exp(16*x) + 9*exp(81*x)). - G. C. Greubel, Mar 11 2023

A094477 Primes of form 2^n + 37^n.

Original entry on oeis.org

2, 1373, 1874177, 23169162752708970943114627382699355445603465075569066753527132965271355336698663708393617779709970177

Offset: 1

Labos Elemer, Jun 01 2004

nonn

The number n must be zero or a power of 2. Checked n being powers of two through 2^20. Thus a(5) > 10^3200000. Primes of this magnitude are rare (about 1 in 7.5 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 29 2013

Cf. A094473, A094474, A082101, A094476.

No more terms for n < 1000, so the next term will be too large to include. - Hugo Pfoertner, Aug 17 2004

A094486 Primes of form 2^j + 223^j.

Original entry on oeis.org

2, 2472973457, 6115597639891380737

Offset: 1

Labos Elemer, Jun 01 2004

Expression 2^j + q^j below q = prime <= prime[130] provided always prime at j=0; or for j=1 if q is a lesser-twin-prime; or more rarely 3 or 4 primes [four ones at q=3,5,17,37,59,137,179,223,461]; never found 5 or more relevant primes and the corresponding exponents proved to be powers of 2. Formal proofs of observations wanted.
See comment by Michael Somos, Aug 27 2004 for proof that j must be zero or a power of 2. - Robert Price, Apr 30 2013
Since the number j must be zero or a power of 2, checked j being powers of two through 2^19. Thus a(5) > 10^2400000. Primes of this magnitude are rare (about 1 in 5.6 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 30 2013

			The relevant exponents are powers of 2: 0,4,8,128. a(4) = 2^128 + 223^128 = 382844.....1067137 (a prime with 301 decimal digits).

Cf. A082101, A094473-A094485.

Corrected by T. D. Noe, Nov 15 2006

A094487 Primes p such that 2^j+p^j are primes for j=0,1,2,4.

Original entry on oeis.org

3, 5, 17, 4517, 5477, 5867, 7457, 8537, 13877, 16067, 22697, 27917, 56477, 59357, 90437, 97577, 101747, 118247, 122207, 124247, 135467, 139457, 140417, 153947, 208697, 247067, 267677, 306947, 419927, 470087, 489407, 520547, 529577, 540347

Offset: 1

Labos Elemer, Jun 01 2004

nonn

			For j=0 1+1=2 is prime; also terms should be lesser-twin-primes
because of p^1+2^1=p+2=prime; 3rd and 4th conditions are as
follows: prime=p^2+4 and prime=16+p^4.

Cf. A082101, A094473-A094486.

{ta=Table[0, {100}], u=1}; Do[s0=2;s1=Prime[j]+2;s2=4+Prime[j]^2;s4=16+Prime[j]^4; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s4], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[45000]],AllTrue[{2+#,4+#^2,16+#^4},PrimeQ]&] (* Harvey P. Dale, Sep 18 2022 *)

A094489 Primes p such that 2^j+p^j are primes for j=0,1,4,32.

Original entry on oeis.org

59, 5417, 19079, 33827, 136949, 181871, 242519, 284897, 421607, 452537, 552401, 598187, 962681, 1068251, 1081979, 1163231, 1317761, 1760279, 1801361, 1891499, 1895081, 1919459, 2056907, 2131601, 2427461, 2557601, 2579177, 2826737

Offset: 1

Labos Elemer, Jun 01 2004

nonn

			For j=0 1+1=2 is prime; also terms should be lesser-twin-primes
because of p^1+2^1=p+2=prime; 3rd and 4th conditions are as
follows: prime=p^4+16 and prime=2^32+p^32.

Harvey P. Dale, Table of n, a(n) for n = 1..400

Cf. A082101, A094473-A094488.

{ta=Table[0, {100}], u=1}; Do[s0=2;s1=Prime[j]+2;s2=4+Prime[j]^2;s8=2^32+Prime[j]^32; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s8], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[210000]],AllTrue[{2+#,16+#^4,2^32+#^32},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 13 2015 *)

cp's OEIS Frontend

A094494 Primes p such that 2^j+p^j are primes for j=0,2,4,8.

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Maple

Mathematica

A082102 Primes of the form 2^(k-1) + 3^k.

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A093988 Numbers k such that 2^k + 3*k is prime.

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Maple

Mathematica

PARI

PFGW

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A094482 Primes of form 2^j + 137^j.

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A161470 Primes of the form 3^k+2^k+k^3-k^2.

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Magma

Mathematica

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A165259 Sum of odd powers of 4 and 9 divided by 13.

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Magma

Mathematica

PARI

SageMath

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A094477 Primes of form 2^n + 37^n.

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A094486 Primes of form 2^j + 223^j.

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A094487 Primes p such that 2^j+p^j are primes for j=0,1,2,4.