A094473 -id:A094473 - cp's OEIS frontend

A045637 Primes of the form p^2 + 4, where p is prime.

13, 29, 53, 173, 293, 1373, 2213, 4493, 5333, 9413, 10613, 18773, 26573, 27893, 37253, 54293, 76733, 85853, 94253, 97973, 100493, 120413, 139133, 214373, 237173, 253013, 299213, 332933, 351653, 368453, 375773, 458333, 552053, 619373

Offset: 1

Felice Russo

These are the only primes that are the sum of two primes squared. 11 = 3^2 + 2 is the only prime of the form p^2 + 2 because all primes greater than 3 can be written as p=6n-1 or p=6n+1, which allows p^2+2 to be factored. - T. D. Noe, May 18 2007
Infinite under the Bunyakovsky conjecture. - Charles R Greathouse IV, Jul 04 2011
All terms > 29 are congruent to 53 mod 120. - Zak Seidov, Nov 06 2013

			29 belongs to the sequence because it equals 5^2 + 4.

T. D. Noe, Table of n, a(n) for n = 1..1000
Yang Ji, Several special cases of a square problem, arXiv:2105.05250 [math.GM], 2021.

The corresponding primes p are in A062324.
Subsequence of A005473 (and thus A185086).
Cf. A094473-A094479.

Select[Prime[ # ]^2+4&/@Range[140], PrimeQ]

forprime(p=2,1e4,if(isprime(t=p^2+4),print1(t","))) \\ Charles R Greathouse IV, Jul 04 2011

a(n) = A062324(n)^2 + 4. - Zak Seidov, Nov 06 2013

Edited by Dean Hickerson, Dec 10 2002

A094475 Primes of form 2^n + 5^n.

Original entry on oeis.org

2, 7, 29, 641

Offset: 1

Labos Elemer, Jun 01 2004

2^n+p^n is prime if n=0;or n=1 and p is a smaller of twin primes; or n=2 and 4+p^2 is prime; or n=3 and 8+p^3 is prime etc. Several conditions have to be satisfied to get a modest number of terms...

n must be zero or a power of two. Checked n being powers of two through 2^22. Thus a(5) > 10^5800000. Primes of this magnitude are rare (about 1 in 13.4 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, May 02 2013

			For n=4, p=2^4+5^4=641, so p can be prime even when the exponent is not a prime.

Cf. A094473, A094474, A082101, A094476.

[ a: n in [0..2100] | IsPrime(a) where a is 5^n+2^n]; // Vincenzo Librandi, Nov 18 2010

Select[Table[2^n+5^n,{n,0,5000}],PrimeQ] (* Harvey P. Dale, May 28 2014 *)

A094476 Primes of form 2^j + 17^j.

Original entry on oeis.org

2, 19, 293, 83537

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) > 10^2500000. Primes of this magnitude are rare (about 1 in 5.9 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+17=19;j=2: p=4+289=293;j=4: p=16+83521=83537; the j exponents are powers of 2.

Cf. A082101, A094473-A094480.

Select[Table[2^n+17^n,{n,0,2000}],PrimeQ] (* Harvey P. Dale, Nov 27 2012 *)

A094488 Primes p such that 2^j+p^j are primes for j=0,1,2,8.

Original entry on oeis.org

137, 2087, 2687, 16067, 24107, 29207, 154787, 155537, 223007, 331907, 427877, 662897, 708137, 769997, 802127, 849047, 869597, 891887, 1031117, 1068497, 1261487, 1336337, 1712567, 1794677, 1807997, 1838297, 1990577, 2189987

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=256+p^8.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A082101, A094473-A094487.

{ta=Table[0, {100}], u=1}; Do[s0=2;s1=Prime[j]+2;s2=4+Prime[j]^2;s8=256+Prime[j]^8; 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[200000]],AllTrue[{#+2,#^2+4,#^8+256},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 03 2018 *)

A094491 Primes p such that 2^j+p^j are primes for j=0,4,8,128.

Original entry on oeis.org

223, 2104547, 2403689, 4268233, 17620457, 21848647, 23487311, 29205821, 42889591, 43458859, 47899487, 48309017, 54666847, 61227457, 73038689, 81742547, 83574457, 85031153, 87285403, 95017003, 100339517, 103136867

Offset: 1

Labos Elemer, Jun 01 2004

nonn

Primes of 2^j+p^j form are a generalization of Fermat-primes. 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-A094490.

			For j=0 1+1=2 is prime; other conditions are: because of p^4+16==prime; 3rd and 4th conditions are as follows: prime=p^8+256 and prime=2^128+p^128.

Cf. A082101, A094473-A094490.

{ta=Table[0, {100}], u=1}; Do[s0=2;s4=16+Prime[j]^4;s8=256+Prime[j]^8;s128=2^128+Prime[j]^128 If[PrimeQ[s0]&&PrimeQ[s4]&&PrimeQ[s8]&&PrimeQ[s128], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]

a(5)-a(22) from Donovan Johnson, Oct 12 2008

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

Original entry on oeis.org

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 *)

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.

A122118 Least prime factor of 2^n + 5^n.

Original entry on oeis.org

2, 7, 29, 7, 641, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 97, 7, 29, 7, 641, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 193, 7, 29, 7, 73, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 97, 7, 29, 7, 641, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 274568286337, 7, 29, 7, 137, 7, 29, 7, 17, 7, 29, 7, 457

Offset: 0

Zak Seidov, Oct 19 2006

nonn

a(n_odd)=7, a(n=2+4k,k=0,1,...)=29, a(64)=274568286337 is unusually large.

Hans Havermann, Table of n, a(n) for n = 0..1023 (terms 0..319 from Antti Karttunen)

Cf. A020639, A074600 (2^n + 5^n), A094475 (primes of form 2^n + 5^n), A122119, A337429.

Cf. also A094473.

Table[FactorInteger[2^n+5^n][[1,1]],{n,0,80}] (* or *) Riffle[Table[ FactorInteger[2^n+5^n][[1,1]],{n,0,80,2}],7] (* The second program is faster *) (* Harvey P. Dale, Mar 02 2015 *)

A122118(n) = { my(k=(2^n+5^n)); forprime(p=if(64==n,274568286337,2),k,if(!(k%p),return(p))); }; \\ Antti Karttunen, Nov 02 2018

a(n) = A020639(A074600(n)). - Antti Karttunen, Nov 02 2018

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

cp's OEIS Frontend

A045637 Primes of the form p^2 + 4, where p is prime.

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A094475 Primes of form 2^n + 5^n.

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Magma

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A094476 Primes of form 2^j + 17^j.

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A094488 Primes p such that 2^j+p^j are primes for j=0,1,2,8.

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A094491 Primes p such that 2^j+p^j are primes for j=0,4,8,128.

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

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

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A122118 Least prime factor of 2^n + 5^n.

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