A062324 -id:A062324 - 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

A157764 Primes p such that p^16 + 2^16 is also prime.

Original entry on oeis.org

89, 107, 127, 139, 173, 179, 229, 233, 349, 421, 461, 521, 557, 571, 727, 863, 991, 1019, 1051, 1069, 1433, 1459, 1627, 1747, 1831, 1877, 2081, 2083, 2591, 2837, 3229, 3319, 3361, 3541, 3677, 3697, 3761, 3877, 4201, 4229, 4259, 4271, 4349, 4451, 4561, 4591, 5011, 5119, 5147, 5171, 5531

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 06 2009

nonn

Primes Q = n^16 + 2^16 only for odd n note: Q is divisible by 97 if n = 97k +- 48, n = 97k +- 50, n = 97k +- 66, n = 97k +- 70, n = 97k +- 78, n = 97k +- 84, n = 97k +- 90, n = 97k +- 92 of course there are similar rules for each prime divisor.

			For n=89: 89^16 + 2^16 = 15496731425178936435099327796097 is prime and 89 is prime too.
For n=3: 3 is (first odd) prime but 3^16 + 2^16 = 43112257 = 3041*14177 (not prime).
For n=85: 85^16 + 2^16 = 7425108623606394726715087956161 is prime too, but 85 is not.

Muniru A Asiru, Table of n, a(n) for n = 1..2915

Cf. A062324.

Filtered(Filtered([1..10^3],IsPrime),p->IsPrime(p) and IsPrime(p^16+2^16)); # Muniru A Asiru, Feb 04 2018

select(p->isprime(p) and isprime(p^16+2^16), [$1..10^4]); # Muniru A Asiru, Feb 04 2018

Select[Prime[Range[800]],PrimeQ[#^16+65536]&] (* Harvey P. Dale, Sep 07 2019 *)

isA157764(n) = isprime(n) && isprime(n^16+65536) \\ Michael B. Porter, Dec 17 2009

More terms from Muniru A Asiru, Feb 05 2018

A116886 Primes p that remain prime through at least 2 iterations of the function f(p) = p^2 + 4.

Original entry on oeis.org

3, 17, 103, 137, 277, 313, 677, 743, 1117, 1627, 2003, 2143, 3407, 3677, 4483, 5087, 5903, 7177, 7333, 8087, 8093, 8147, 8537, 8573, 9293, 9473, 10177, 10477, 11173, 13807, 14897, 15107, 16657, 19753, 21563, 22307, 24113, 26113, 26417, 26633

Offset: 1

Giovanni Resta, Feb 27 2006

nonn

Numbers p with the property that p, q = p^2 + 4, and r = q^2 + 4 are all prime. - Zak Seidov, Sep 08 2009

a(n) = sqrt(A165218(n) - 4). - Zak Seidov, Sep 08 2009

			17 is prime, 17^2 + 4 = 293 is prime and 293^2 + 4 = 85853 is prime.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A062324, A116887, A116888, A116889, A045637, A062324, A165218.

Select[Prime[Range[2*7! ]],PrimeQ[ #^2+4]&&PrimeQ[(#^2+4)^2+4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
fQ[n_]:=AllTrue[Rest[NestList[#^2+4&,n,2]],PrimeQ]; Select[Prime[ Range[ 3000]],fQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 21 2014 *)

is(n)=my(q);isprime(p) && isprime(q=p^2+4) && isprime(q^2+4) \\ Charles R Greathouse IV, Nov 06 2013

Edited by N. J. A. Sloane, Sep 18 2009 at the suggestion of R. J. Mathar

A118940 Primes p such that (p^2+7)/8 is prime.

Original entry on oeis.org

3, 7, 17, 23, 41, 47, 71, 89, 103, 113, 127, 137, 151, 191, 193, 199, 223, 263, 271, 281, 337, 359, 401, 439, 457, 503, 521, 569, 577, 599, 641, 719, 727, 751, 839, 857, 863, 881, 887, 929, 991, 1009, 1033, 1097, 1103, 1151, 1193, 1217, 1231, 1279, 1297, 1303

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 8 divides q^2+7.

Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+7)/8]&]

lista(nn) = {forprime(p=2, nn, iferr(if (isprime(q=(p^2+7)/8), print1(q, ", ")), E, ););} \\ Michel Marcus, Feb 18 2018

A176983 Primes p such that smallest prime q > p^2 is of form q = p^2 + k^2.

Original entry on oeis.org

2, 5, 7, 13, 17, 37, 47, 67, 73, 97, 103, 137, 163, 167, 193, 233, 277, 281, 293, 307, 313, 317, 347, 373, 389, 421, 439, 461, 463, 487, 499, 503, 547, 571, 577, 593, 607, 613, 661, 677, 691, 701, 739, 743, 769, 787, 821, 823, 827, 829, 853, 883, 953, 967, 983

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2010

nonn

By Fermat's 4n+1 theorem, q must be congruent to 1 (mod 4).

Corresponding values of k: 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 4, 6, 2, 2, 4, 2. - Zak Seidov, Nov 04 2013

			17 is here because 293 is the first prime after 17^2 and 293 = 17^2 + 2^2.

Eric W. Weisstein, Fermat's 4n+1 Theorem

Cf. A000040, A000290, A002144, A159828.

A062324 is subsequence. - Zak Seidov, Nov 04 2013

Select[Prime[Range[200]], IntegerQ[Sqrt[NextPrime[ #^2] - #^2]] & ]

Edited and extended by T. D. Noe, May 12 2010

A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.

Original entry on oeis.org

9, 11, 19, 21, 23, 25, 29, 31, 39, 41, 43, 49, 51, 53, 55, 59, 61, 63, 69, 71, 75, 77, 79, 81, 83, 89, 91, 93, 99, 101, 105, 107, 109, 111, 113, 119, 121, 123, 127, 129, 131, 133, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161, 165, 169, 171, 173, 175, 179, 181, 185, 187, 189, 191, 195, 197, 199

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

nonn

Conjecture: All terms are odd. An equivalent conjecture in A263977 is that if k > 0 is even, then k^2 + p^2 is prime for some prime p. We have checked this for all k <= 12*10^7.
An odd number k is a term if and only if k^2 + 2^2 is composite, since k^2 + p^2 is even and > 2 (hence composite) for all odd primes p. Thus if the Conjecture is true, then the sequence is simply odd k such that k^2 + 4 is composite. The sequence of odd k, such that k^2 + 4 is prime, is A007591.
The complementary sequence is A263977. Given k in it, the smallest prime p, such that k^2 + p^2 is prime, is in A263726. These numbers k^2 + p^2 form A185086, the Fouvry-Iwaniec primes.

			9^2 + 2^2 = 85 = 5*17, 11^2 + 2^2 = 125 = 5^3, and 23^2 + 2^2 = 533 = 13*41 are composite, so 9, 11, and 23 are members.
1^2 + 2^2 = 5 and 2^2 + 3^3 = 13 are prime, so 1, 2, and 3 are not members.

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, arXiv:math/0703284 [math.NT], 2007-2008; Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A002313, A045637, A062324, A185086, A007591, A240130, A240131, A263466, A263726, A263977.

A116887 Primes p that remain prime through at least 3 iterations of function f(p)=p^2+4.

Original entry on oeis.org

5087, 11173, 16657, 47017, 54503, 185243, 207643, 300413, 306167, 341813, 391387, 849047, 1348577, 1438223, 1502923, 1857407, 1909267, 2121737, 2161163, 2288773, 2610133, 2725157, 2744723, 2779097, 2874463, 2881327, 3079927, 3149077, 3154483, 3173683, 3194483

Offset: 1

Giovanni Resta, Feb 27 2006

nonn

			p=5087, f(p)= 25877573, f(f(p))= 669648784370333 and f(f(f(p)))= 448429494408664742387290530893 are all primes.

Zak Seidov, Table of n, a(n) for n = 1..2567 (all terms up to 10^9).

Cf. A062324, A116886, A116888, A116889.

Select[Prime[Range[8! ]],PrimeQ[ #^2+4]&&PrimeQ[(#^2+4)^2+4]&&PrimeQ[ ((#^2+4)^2+4)^2+4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)

A116889 a(n) is the least prime p that remains prime through n iterations of function f(p)=p^2+4.

Original entry on oeis.org

2, 3, 3, 5087, 306167

Offset: 0

Giovanni Resta, Feb 27 2006

The sequence is finite, since it can be proved that if p, f(p), f(f(p)), f(f(f(p))) and f(f(f(f(p)))) are all primes, then the next iteration gives a multiple of 13, greater than 13, thus a(k) for k>=5 does not exist.

			a(0)=2 since f(2)=8 is not prime. a(1)=a(2)=3 since both f(3)=13 and f(f(3))=173 are primes.

Cf. A062324, A116886, A116887, A116888.

Typo in Example fixed by Zak Seidov, Nov 07 2013

A118939 Primes p such that (p^2+3)/4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 29, 31, 41, 43, 67, 83, 101, 109, 139, 151, 157, 179, 181, 199, 211, 223, 239, 263, 277, 283, 307, 311, 337, 347, 353, 379, 389, 419, 431, 463, 491, 557, 577, 587, 619, 659, 673, 739, 757, 797, 809, 811, 829, 853, 907, 911, 953, 991, 1051

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2+3. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118940, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+3)/4]&]

A158361 Primes p with property that Q = p^4 + 2^4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 37, 41, 59, 61, 71, 79, 97, 131, 139, 179, 223, 227, 229, 241, 283, 313, 317, 359, 367, 379, 383, 389, 439, 449, 461, 487, 503, 521, 569, 593, 617, 619, 631, 661, 683, 709, 733, 811, 821, 853, 911, 977, 1049, 1061, 1063, 1069, 1091, 1093, 1117

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2009

Q is always congruent to 1 (mod 4).
Q is divisible by 17 if p is congruent to 1, 4, 13, or 16 (mod 17).
It is conjectured that sequence a(n) is infinite.
Q is in A094479. - Zak Seidov, Jul 08 2020

			3 is in the sequence since for p=3: p^4+2^4 = 3^4+16 = 97 is prime.
29 is not in the sequence since 29^4+2^4 = 707297 = 73 x 9689 is not prime.

Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
Richard Guy, "Unsolved Problems in Number Theory"

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A062324, A094479, A157950, A157764.

[p: p in PrimesUpTo(2000) | IsPrime(p^4+16)]; // Vincenzo Librandi, Jun 18 2014

Select[Range[10^3], PrimeQ[#] && PrimeQ[#^4 + 16] &] (* Vincenzo Librandi, Jun 18 2014 *)
Select[Prime[Range[200]],PrimeQ[#^4+16]&] (* Harvey P. Dale, Jun 23 2014 *)

isA158361(n) = isprime(n) && isprime(n^4+16)

Corrected and edited by Michael B. Porter, Dec 17 2009

cp's OEIS Frontend

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

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A157764 Primes p such that p^16 + 2^16 is also prime.

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A116886 Primes p that remain prime through at least 2 iterations of the function f(p) = p^2 + 4.

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A118940 Primes p such that (p^2+7)/8 is prime.

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A176983 Primes p such that smallest prime q > p^2 is of form q = p^2 + k^2.

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A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.

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A116887 Primes p that remain prime through at least 3 iterations of function f(p)=p^2+4.

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A116889 a(n) is the least prime p that remains prime through n iterations of function f(p)=p^2+4.

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