A045637 -id:A045637 - cp's OEIS frontend

A094481 Duplicate of A045637.

13, 29, 53, 173, 293, 1373, 2213, 4493, 5333, 9413, 10613, 18773, 26573, 27893

Offset: 1

dead

A062324 Primes p such that p^2 + 4 is also prime.

3, 5, 7, 13, 17, 37, 47, 67, 73, 97, 103, 137, 163, 167, 193, 233, 277, 293, 307, 313, 317, 347, 373, 463, 487, 503, 547, 577, 593, 607, 613, 677, 743, 787, 823, 827, 853, 883, 953, 967, 983, 997, 1087, 1117, 1123, 1237, 1367, 1423, 1447, 1523, 1543, 1613

Offset: 1

Reiner Martin, Jul 12 2001

Equivalent to the definition: largest absolute dimension of Gaussian primes with prime coordinates. As 2 is the only even prime, the only possibility for a Gaussian prime to have prime coordinates is to be of the form +/-2 +/- I*p or +/-p +/-2*I with p^2+4 a prime, i.e., p is a member of this sequence. - Olivier Gérard, Aug 17 2013
When p > 3, p^2 + 2 is never prime. - Zak Seidov, Nov 04 2013
For p > 5 and q = p^2 + 4, the following congruences apply: q == 3 (mod 10) and q == 5 (mod 12). - Joseph Wheat, Feb 28 2025

			a(1) = 3 because 3^2 + 4 = 13 is prime,
a(4) = 13 because 13^2 + 4 = 173 is prime. - _Zak Seidov_, Nov 04 2013

Harry J. Smith, 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^2+4 are in A045637.

Subsequence of A176983.

Select[Prime/@Range[300], PrimeQ[ #^2+4]&]

{ n=0; forprime (p=2, 5*10^5, if (isprime(p^2 + 4), write("b062324.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 04 2009

a(n) = sqrt(A045637(n) - 4). - Zak Seidov, Nov 04 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2001

Edited by Dean Hickerson, Dec 10 2002

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

A182475 Primes of the form p^2+10, where p is prime.

Original entry on oeis.org

19, 59, 131, 179, 971, 2819, 3491, 5051, 6899, 9419, 10211, 16139, 22811, 24659, 32051, 32771, 44531, 49739, 51539, 57131, 96731, 134699, 143651, 201611, 237179, 271451, 358811, 361211, 375779, 383171, 398171, 552059, 597539, 654491, 683939, 779699, 954539

Offset: 1

Alex Ratushnyak, May 01 2012

This is also the sequence of prime numbers expressible as p1^2+p2^2+1 where p1 and p2 are also prime - Christian N. K. Anderson, Mar 25 2013

Cf. A045637 (p^2 + 4 is prime), A079141 (p^2 + 6 is prime), A138355.

Select[Table[p^2 + 10, {p, Prime[Range[200]]}], PrimeQ] (* T. D. Noe, May 01 2012 *)

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.

A165218 Primes q of the form q=p^2+4 (p=prime) such that r=q^2+4 is also prime.

Original entry on oeis.org

13, 293, 10613, 18773, 76733, 97973, 458333, 552053, 1247693, 2647133, 4012013, 4592453, 11607653, 13520333, 20097293, 25877573, 34845413, 51509333, 53772893, 65399573, 65496653, 66373613, 72880373, 73496333, 86359853, 89737733

Offset: 1

Zak Seidov, Sep 08 2009

nonn

Intersection of A062324 and A045637. Except of the first term, 13, all terms == 5 (mod 6) == 5 (mod 12) == 5 (mod 24) == 23 (mod 30)== 53 (mod 120). Values of primes p in A116886.

			Prime q=13=p^2+4 (p=3) and r=q^2+4=13^2+4=173 (prime).
Prime q=293=p^2+4 (p=17) and r=q^2+4=293^2+4=85853 (prime).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A045637, A062324, A116886.

Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[PrimeQ[q = p^2+4] && PrimeQ[q^2+4], Print[q]; Sow[q]]]][[2, 1]] (* Jean-François Alcover, Nov 07 2013 *)
Select[Prime[Range[2000]]^2+4,AllTrue[{#,#^2+4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2018 *)

a(n) = (A116886(n))^2 + 4.

A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

Original entry on oeis.org

2, 3, 5, 5, 7, 5, 3, 5, 3, 7, 11, 7, 11, 13, 7, 2, 13, 13, 5, 17, 13, 11, 5, 17, 7, 17, 19, 3, 17, 7, 19, 5, 11, 19, 13, 23, 5, 17, 19, 13, 23, 5, 2, 19, 17, 11, 5, 23, 29, 29, 23, 19, 29, 13, 31, 31, 23, 11, 3, 5, 31, 13, 2, 29, 5, 13, 31, 2, 11, 5, 31, 37, 23, 37, 3, 7, 23, 3, 13, 31, 19, 37, 41, 11

Offset: 1

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

nonn

The sequence is well-defined by the uniqueness part of Fermat's two-squares theorem.

The sequence is infinite, since Fouvry and Iwaniec proved that A185086 is infinite.

			A185086(2) = 13 = 2^2 + 3^2 and A185086(6) = 61 = 5^2 + 6^2, so a(2) = 3 and a(6) = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A002313, A002144, A028916, A045637, A062324, A185086, A240130, A262340, A263722.

p = 2; lst = {}; While[p < 100, k = 1; While[k < 101, If[PrimeQ[k^2 + p^2], AppendTo[lst, {k^2 + p^2, If[PrimeQ@ k, Max[k, p], p]}]]; k++]; p = NextPrime@ p]; Transpose[Union@ lst][[2]]

do(lim)=my(v=List(),p2,t); forprime(p=2, sqrtint(lim\=1), p2=p^2; for(k=1, sqrtint(lim-p2), if(isprime(t=p2+k^2), listput(v, [t, if(isprime(k),max(k,p),p)])))); v=vecsort(Set(v),1); apply(u->u[2], v) \\ Charles R Greathouse IV, Aug 21 2017

a(n)^2 = A185086(n) - k^2 for some integer k > 0.

A092104 Primes of form p*q + 4, with prime p and q.

Original entry on oeis.org

13, 19, 29, 37, 43, 53, 59, 61, 73, 89, 97, 127, 137, 149, 163, 173, 181, 191, 223, 239, 241, 251, 257, 263, 269, 271, 293, 307, 313, 331, 359, 397, 419, 421, 431, 449, 457, 509, 521, 523, 541, 547, 557, 563, 569, 577, 587, 593, 601, 653, 659, 673, 683, 691

Offset: 1

Zak Seidov, Feb 20 2004

Primes of form p*q + 2, A063638. Primes common in A063638 and A092104, A092105. Primes of form p*p + 4, A045637.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A045637, A063638, A092105.

With[{upto=700},Select[Times@@#+4&/@Tuples[Prime[Range[PrimePi[upto/2]]], 2], PrimeQ[#]&&#<+upto&]]//Union (* Harvey P. Dale, Jul 23 2016 *)

list(lim)=my(v=List(),t); forprime(p=3,(lim-4)\3, forprime(q=3,min((lim-4)\p, p), t=p*q+4; if(isprime(t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 05 2016

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.

A244637 Primes that are the sum of the squares of distinct primes.

Original entry on oeis.org

13, 29, 53, 83, 173, 179, 199, 227, 293, 347, 367, 373, 419, 439, 463, 467, 487, 491, 541, 563, 569, 587, 607, 613, 617, 641, 653, 659, 709, 727, 733, 751, 809, 823, 827, 829, 853, 857, 877, 881, 919, 953, 971, 977, 991, 997, 1013, 1019, 1021, 1039, 1049

Offset: 1

Michel Marcus, Jul 03 2014

nonn

Primes in A048261.
Provide the prime factors of A185077.
A045637 is a subsequence.
There are only 368 primes not in this sequence, the largest being 12601. - Robert Israel, Jul 04 2014

			13 is in the sequence since it is prime and 13 = 2^2 + 3^2 (2 and 3 are distinct primes).

Vincenzo Librandi, Table of n, a(n) for n = 1..2870

Cf. A045637, A048261, A121518, A185077.

nn=10;s={0};Do[p=Prime[n];s=Union[s,s+p^2],{n,nn}];Select[s,(0<#<=Prime[nn]^2)&&PrimeQ[#]&] (* Michel Lagneau, Jul 03 2014 *)

cp's OEIS Frontend

A094481 Duplicate of A045637.

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A062324 Primes p such that p^2 + 4 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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A182475 Primes of the form p^2+10, where p is prime.

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

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A165218 Primes q of the form q=p^2+4 (p=prime) such that r=q^2+4 is also prime.

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A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

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A092104 Primes of form p*q + 4, with prime p and q.

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