A240130 -id:A240130 - cp's OEIS frontend

A185086 Fouvry-Iwaniec primes: Primes of the form k^2 + p^2 where p is a prime.

5, 13, 29, 41, 53, 61, 73, 89, 109, 113, 137, 149, 157, 173, 193, 229, 233, 269, 281, 293, 313, 317, 349, 353, 373, 389, 397, 409, 433, 449, 461, 509, 521, 557, 569, 593, 601, 613, 617, 653, 673, 701, 733, 761, 773, 797, 809, 853, 857, 877, 929, 937, 941, 953

Offset: 1

Charles R Greathouse IV, Feb 18 2011

Sequence is infinite, see Fouvry & Iwaniec.
Its intersection with A028916 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015
Named after the French mathematician Étienne Fouvry (b. 1953) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 20 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica, Vol. 79, No. 3 (1997), pp. 249-287.
Lasse Grimmelt, Vinogradov's Theorem with Fouvry-Iwaniec Primes, arXiv:1809.10008 [math.NT], 2018.
Art of Problem Solving, Fermat's Two Squares Theorem.

Subsequence of A002144 and hence of A002313.
The positive terms of A240130 form a subsequence.
Cf. A010052, A001248, A000040, A028916, A262340.

a185086 n = a185086_list !! (n-1)
a185086_list = filter (\p -> any ((== 1) . a010052) $
               map (p -) $ takeWhile (<= p) a001248_list) a000040_list
-- Reinhard Zumkeller, Mar 17 2013

nn = 1000; Union[Reap[Do[n = k^2 + p^2; If[n <= nn && PrimeQ[n], Sow[n]], {k, Sqrt[nn]}, {p, Prime[Range[PrimePi[Sqrt[nn]]]]}]][[2, 1]]]

is(n)=forprime(p=2,sqrtint(n),if(issquare(n-p^2),return(isprime(n))));0

list(lim)=my(v=List(),N,t);forprime(p=2,sqrt(lim), N=p^2; for(n=1,sqrt(lim-N), if(ispseudoprime(t=N+n^2), listput(v,t)))); v=vecsort(Vec(v),,8); v

A240131 Least k such that prime(n)^2 + k^2 is prime, or 0 if none.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 6, 8, 4, 4, 2, 4, 8, 2, 10, 6, 10, 2, 6, 2, 4, 18, 4, 2, 10, 2, 10, 4, 18, 8, 16, 2, 10, 14, 4, 10, 2, 2, 10, 4, 6, 4, 2, 8, 16, 4, 18, 8, 4, 2, 10, 16, 14, 18, 8, 10, 6, 2, 4, 8, 2, 2, 4, 2, 2, 6, 20, 2, 14, 8, 10, 8, 2, 6, 12, 4, 18, 4, 6, 14, 4, 6, 12, 4, 28, 10, 12, 6, 2, 12, 14, 2, 6, 4, 2, 14, 14, 10, 6

Offset: 1

Jonathan Sondow, Apr 07 2014

nonn

The main entry for this sequence is A240130.
k and p_n must be of opposite parity. Conjecture, there is always a k for any p_n. Tested for all primes < 15*10^10. - Robert G. Wilson v, Nov 04 2015
Least k > 0 for which prime(n)+i*k is a Gaussian prime. - Robert Israel, Nov 04 2015

			Prime(3) = 5 and 5^2 + 1^2 = 26 is not prime but 5^2 + 2^2 = 29 is prime, so a(3) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A240130.

f:= proc(p) local k; for k from 2 by 2 do if isprime(p^2 + k^2) then return k fi eod nd proc:
f(2):= 1:
map(f, select(isprime, [2,seq(2*i+1,i=1..10000)])); # Robert Israel, Nov 04 2015

f[n_] := Block[{k = If[n == 1, 1, 2], p = Prime@ n}, While[ !PrimeQ[k^2 + p^2], k += 2]; k]; Array[f, 100]  (* Robert G. Wilson v, Nov 03 2015 *)
lk[n_]:=Module[{k=2,n2=n^2},While[!PrimeQ[n2+k^2],k+=2];k]; Join[{1}, Table[ lk[x],{x,Prime[Range[2,100]]}]] (* Harvey P. Dale, Mar 22 2019 *)

vector(100, n, p = prime(n); k = 1 - p%2; inc = 2; while (!isprime(q=p^2+k^2), k += inc); k; ) \\ Altug Alkan, Nov 04 2015

a(n)^2 = A240130(n) - prime(n)^2 if a(n) > 0.

A263466 Least k such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, or 0 if none.

Original entry on oeis.org

1, 2, 4, 12, 14, 48, 178, 44, 152, 66, 224, 272, 496, 322, 408, 2068, 114, 354, 592, 584, 3192, 406, 2708, 774, 2658, 394, 4102, 2432, 3346, 2562, 8722, 4424, 9562, 2986, 6856, 1714, 21318, 5858, 7568, 16272, 7576, 4864, 6244, 29262, 29992, 9996, 10406, 58348, 16872, 11384, 12738, 22126, 9946, 24214, 81682, 46082, 74616, 88016, 6788, 30856, 21542, 38672, 131492, 62874, 75358, 95262, 39554, 83552, 65022, 73664

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Nov 02 2015

nonn

The data support the conjecture in A263977 that if k > 0 is even, then k^2 + p^2 is prime for some prime p.

a(n) is the location of the first occurrence of prime(n) in A263978.

			The primes p < prime(3) = 5 are p = 2 and 3. As 1^2 + 2^2 = 5, 2^2 + 3^2 = 13, and 3^2 + 2^2 = 13 are prime, a(3) >= 4. But 4^2 + 2^2 = 20 and 4^2 + 3^2 = 25 are not prime, while 4^2 + 5^2 = 41 is prime, so a(3) = 4.

Jonathan Sondow and Robert G. Wilson v, Table of n, a(n) for n = 1..260
Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, 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. A240130, A240131, A263721, A263722, A263726, A263977, A263978.

f[n_] := Block[{p = 2}, While[ !PrimeQ[n^2 + p^2], p = NextPrime@ p]; p]; t = 0*Range@ 300; t[[1]] = 1; k = 2; While[k < 50000001, p = f@ k; If[ t[[PrimePi@ p]] == 0, t[[PrimePi@ p]] = k; Print[{PrimePi@ p, p, k}]]; k += 2]; t

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.

A263726 Least prime p such that p^2 + A263977(n)^2 is prime.

Original entry on oeis.org

2, 3, 2, 5, 2, 5, 2, 3, 3, 7, 2, 11, 2, 5, 2, 5, 3, 5, 5, 5, 2, 5, 11, 3, 2, 5, 2, 5, 2, 3, 3, 5, 19, 2, 5, 2, 13, 7, 3, 11, 11, 2, 3, 13, 3, 11, 2, 29, 2, 5, 3, 5, 2, 5, 5, 7, 7, 3, 11, 2, 11, 2, 3, 11, 7, 5, 2, 5, 2, 3, 3, 5, 2, 11, 5, 5, 3, 3, 59, 2, 11, 2, 3, 7, 13, 5, 2, 5, 7

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 30 2015

nonn

The least k, such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, is in A263466.

			A263977(1) = 1, and 2 and 2^2 + 1^2 = 5 are prime, so a(1) = 2.

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, 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. A240130, A240131, A263466, A263722, A263977.

f[n_] := Block[{p = 2}, While[! PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@ p]; If[p > 1500, 0, p]]; lst = {}; k = 1; While[k < 130, If[f@ k > 0, AppendTo[lst, f@ k]]; k++]; lst

A263977 Integers k > 0 such that k^2 + p^2 is prime for some prime p.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 60, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 94, 95, 96, 97, 98, 100, 102, 103, 104, 106, 108, 110, 112, 114, 115, 116, 117, 118, 120, 122, 124, 125, 126

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 30 2015

nonn

The smallest such prime p is in A263726.
Complement of A263722.
An odd number k is a member if and only if k^2 + 4 is prime; see A007591.
Conjecture: Every even number k is a member. (This is equivalent to the Conjecture in A263722.) We have checked this for all k <= 12*10^7.

			1^2 + 2^2 = 5, and 2 and 5 are prime, so a(1) = 1.
9^2 + p^2 is composite for all primes p, so 9 is not a member.

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, 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. A007591, A240130, A240131, A263466, A263722, A263726.

fQ[n_] := Block[{p = 2}, While[ !PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@ p]; If[p > 1500, 0, p]]; lst = {}; k = 1; While[k < 130, If[fQ@ k > 0, AppendTo[lst, k]]; k++]; lst

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.

A263978 Least prime p such that n^2 + p^2 is prime, or 0 if none.

Original entry on oeis.org

2, 3, 2, 5, 2, 5, 2, 3, 0, 3, 0, 7, 2, 11, 2, 5, 2, 5, 0, 3, 0, 5, 0, 5, 0, 5, 2, 5, 0, 11, 0, 3, 2, 5, 2, 5, 2, 3, 0, 3, 0, 5, 0, 19, 2, 5, 2, 13, 0, 7, 0, 3, 0, 11, 0, 11, 2, 3, 0, 13, 0, 3, 0, 11, 2, 29, 2, 5, 0, 3, 0, 5, 2, 5, 0, 5, 0, 7, 0, 7, 0, 3, 0, 11, 2, 11, 2, 3, 0, 11, 0, 7, 0, 5, 2, 5, 2, 3, 0, 3

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Nov 06 2015

nonn

When n is odd, n^2 + p^2 is composite for all odd primes p, so a(n) = 2 or 0 according as n^2 + 2^2 is prime or not.
The locations of the zeros are in A263722.
The location of the first occurrence of prime(n) is A263466(n).

			a(1) = 2 since 1^2 + 2^2 = 5 is prime.
a(2) = 3 since 2^2 + 2^2 = 8 is not prime but 2^2 + 3^2 = 13 is prime.
a(9) = 0 since 9^2 + 2^2 = 85 is not prime.

Cf. A240130, A240131, A263466, A263721, A263722, A263726, A263977.

f[n_] := If[OddQ[n] && ! PrimeQ[n^2 + 4], 0,
  Block[{p = 2}, While[! PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@p];
   p]]; Array[f, 100]

cp's OEIS Frontend

A185086 Fouvry-Iwaniec primes: Primes of the form k^2 + p^2 where p is a prime.

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A240131 Least k such that prime(n)^2 + k^2 is prime, or 0 if none.

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A263466 Least k such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, or 0 if none.

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

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A263726 Least prime p such that p^2 + A263977(n)^2 is prime.

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A263977 Integers k > 0 such that k^2 + p^2 is prime for some prime p.

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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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A263978 Least prime p such that n^2 + p^2 is prime, or 0 if none.

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