A007519 -id:A007519 - cp's OEIS frontend

A228228 Primes congruent to {3, 5, 13, 15} mod 16.

3, 5, 13, 19, 29, 31, 37, 47, 53, 61, 67, 79, 83, 101, 109, 127, 131, 149, 157, 163, 173, 179, 181, 191, 197, 211, 223, 227, 229, 239, 269, 271, 277, 293, 307, 317, 349, 367, 373, 383, 389, 397, 419, 421, 431, 461, 463, 467, 479, 499, 509, 541, 547, 557, 563

Offset: 1

Arkadiusz Wesolowski, Aug 16 2013

nonn

Union of A091968, A127589, A141196, and A127576.

Let p be a prime number and let E(p) denote the elliptic curve y^2 = x^3 + p*x. If p is in the sequence, then the rank of E(p) is 0 or 1. Therefore, A060953(a(n)) must be one of only two values: 0 or 1.

J. H. Silverman, The arithmetic of elliptic curves, Springer, NY, 1986, p. 311.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A007519, A060953, A091968, A127576, A127589, A141196, A228227.

[p: p in PrimesUpTo(563) | p mod 16 in {3, 5, 13, 15}];

Select[Prime@Range[103], MemberQ[{3, 5, 13, 15}, Mod[#, 16]] &]

A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

Original entry on oeis.org

7, 5, 7, 17, 13, 29, 19, 41, 73, 31, 97, 191, 43, 89, 97, 109, 61, 311, 137, 73, 149, 241, 337, 181, 197, 103, 313, 109, 331, 229, 257, 397, 139, 281, 151, 457, 317, 821, 337, 349, 181, 547, 193, 389, 199, 401, 1061, 449, 229, 461, 937, 241, 727, 757, 1033, 1321, 271, 1361, 557

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2015

nonn

Least prime q such that q == 1 (mod prime(n) + 1).

Consists of the initial terms of sequences A002476, A002144, A002476, A007519, A068228, A140444, A061237, A141881, A107008, A132230, A133870, A141868, A124826, A142292, A142398, A141948, A088955, A142003, ...

Cf. A263729, A263730, A263769.

Table[q = 2; While[! IntegerQ[(Prime[n]^2 + q Prime@ n)/(Prime@ n + 1)], q = NextPrime@ q]; q, {n, 59}] (* Michael De Vlieger, Oct 26 2015 *)

a(n) = {p = prime(n); q = 2; while ((p^2 + p*q) % (p + 1), q = nextprime(q+1)); q;} \\ Michel Marcus, Oct 26 2015

5 is in this sequence because (prime(2)^2 + 5*prime(2))/(prime(2) + 1) = 6 and 5 is prime.

A269424 Record (maximal) gaps between primes of the form 8k + 1.

Original entry on oeis.org

24, 32, 56, 64, 88, 112, 120, 136, 160, 216, 232, 240, 264, 304, 384, 480, 488, 528, 544, 576, 624, 640, 720, 760, 816, 888, 960, 1032, 1064, 1200, 1296, 1320, 1432, 1464, 1520, 1560, 1608, 1832, 1848

Offset: 1

Alexei Kourbatov, Feb 25 2016

nonn

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 1 below x are about phi(8)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(8)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(8)=4.
Conjecture: a(n) < phi(8)*log^2(A269426(n)) almost always.
A269425 lists the primes preceding the maximal gaps.
A269426 lists the corresponding primes at the end of the maximal gaps.

			The first two primes of the form 8k + 1 are 17 and 41, so a(1)=41-17=24. The next prime of this form is 73 and the gap 73-41=32 is a new record, so a(2)=32.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007519, A269425, A269426.

re = 0; s = 17; Reap[For[p = 41, p < 10^8, p = NextPrime[p], If[Mod[p, 8] == 1, g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]]][[2, 1]] (* Jean-François Alcover, Oct 17 2016, adapted from PARI *)

re=0; s=17; forprime(p=41, 1e8, if(p%8!=1, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

A269425 Primes 8k + 1 preceding the maximal gaps in A269424.

Original entry on oeis.org

17, 41, 137, 457, 673, 2161, 5953, 8377, 10009, 22481, 37657, 73121, 79889, 220897, 351529, 1879121, 2321393, 4259113, 6394657, 8211977, 9618457, 11282017, 36087113, 59502217, 72495233, 236885513, 556952881, 809097481, 830449097, 888023449, 2420630497, 3845315977, 13243532017, 17279668529, 29704277129, 49624608961, 59974490209, 107046775289, 158191299481

Offset: 1

Alexei Kourbatov, Feb 25 2016

nonn

Subsequence of A007519.

A269424 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 8k + 1 are 17 and 41, so a(1)=17. The next prime of this form is 73 and the gap 73-41=32 is a new record, so a(2)=41.

Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007519, A269424, A269426.

re=0; s=17; forprime(p=41, 1e8, if(p%8!=1, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

A269426 Primes 8k + 1 at the end of the maximal gaps in A269424.

Original entry on oeis.org

41, 73, 193, 521, 761, 2273, 6073, 8513, 10169, 22697, 37889, 73361, 80153, 221201, 351913, 1879601, 2321881, 4259641, 6395201, 8212553, 9619081, 11282657, 36087833, 59502977, 72496049, 236886401, 556953841, 809098513, 830450161, 888024649, 2420631793, 3845317297, 13243533449, 17279669993, 29704278649, 49624610521, 59974491817, 107046777121, 158191301329

Offset: 1

Alexei Kourbatov, Feb 25 2016

nonn

Subsequence of A007519.

A269424 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 8k + 1 are 17 and 41, so a(1)=41. The next prime of this form is 73 and the gap 73-41=32 is a new record, so a(2)=73.

Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007519, A269424, A269425.

re=0; s=17; forprime(p=41, 1e8, if(p%8!=1, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

A022761 n-th 8k+1 prime plus n-th 8k+7 prime.

Original entry on oeis.org

24, 64, 104, 136, 168, 192, 240, 320, 384, 408, 448, 480, 536, 576, 616, 672, 720, 792, 816, 840, 952, 1008, 1040, 1072, 1088, 1120, 1240, 1280, 1392, 1416, 1528, 1584, 1624, 1680, 1760, 1792, 1840, 1896, 1944, 1968, 2064, 2112, 2144, 2224

Offset: 1

Clark Kimberling

nonn

			The first four primes of the form 8k - 1 are 7, 23, 31, 47. The first four primes of the form 8k + 1 are 17, 41, 73, 89.
Thus a(1) = 7 + 17  = 24.
a(2) = 23 + 41 = 64.
a(3) = 31 + 73 = 104.
a(4) = 47 + 89 = 136.

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

Cf. A007522, A007519.

thresh = 100; A007522 = Select[8Range[thresh] - 1, PrimeQ]; A007519 = Select[8Range[thresh] + 1, PrimeQ]; preExh = Min[Length[A007522], Length[A007519]]; Take[A007522, preExh] + Take[A007519, preExh]
Module[{nn=300,p1,p7,len},p1=Select[Prime[Range[nn]],IntegerQ[(#-1)/8]&];p7=Select[Prime[Range[nn]],IntegerQ[(#-7)/8]&];len=Min[ Length[ p1],Length[ p7]];Total/@Thread[{Take[p1,len],Take[p7,len]}]] (* Harvey P. Dale, May 26 2020 *)

A121243 Primes of the form 4x^2 + 4xy + 9y^2.

Original entry on oeis.org

17, 73, 89, 97, 193, 233, 241, 281, 401, 433, 449, 601, 617, 641, 673, 769, 929, 937, 977, 1009, 1033, 1049, 1097, 1193, 1289, 1297, 1361, 1409, 1433, 1481, 1489, 1609, 1697, 1721, 1753, 1801, 1873, 1913

Offset: 1

Steven Finch, Aug 22 2006

nonn

This sequence is complementary to A105389 in the sense that the two sequences are disjoint and their union constitutes all primes p satisfying Mod[p,8]=1.

Primes satisfying Mod[p,8]=1 are of form x^2+8y^2 (A007519), with the sequence above as odd y, while A105389 is even y. This can be seen by expressing the former as (2x+y)^2+8y^2 (where y can only be odd), while the latter is u^2+8(2v)^2. [From Tito Piezas III, Jan 01 2009]

			17 = 4*1^2 + 4*1*1 + 9*1^2, 73 = 4*1^2 + 4*1*(-3) + 9*(-3)^2

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Claudio Qureshi, Antonio Campello, Sueli I. R. Costa, Non-Existence of Linear Perfect Lee Codes With Radius 2 for Infinitely Many Dimensions, IEEE Transactions on Information Theory (2018) Vol. 64, Issue 4, pp. 3042-3047.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A105389.

QuadPrimes2[4, -4, 9, 10000] (* see A106856 *)
(* Second program: *)
max = 2000; Table[yy = {y, Floor[-2x/9 - 1/9 Sqrt[9max - 32x^2]], Ceiling[-2x/9 + 1/9 Sqrt[9max - 32x^2]]}; Table[4x^2 + 4 x y + 9y^2, yy // Evaluate], {x, 0, Ceiling[3Sqrt[max]/(4Sqrt[2])]}] // Flatten // Union // Select[#, # <= max && PrimeQ[#]&]& // Quiet (* Jean-François Alcover, Oct 08 2018 *)

A139493 Primes of the form x^2 + 9x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

11, 23, 37, 53, 67, 71, 113, 137, 163, 179, 191, 317, 331, 379, 389, 401, 421, 443, 449, 463, 487, 499, 599, 617, 631, 641, 653, 683, 709, 751, 757, 823, 863, 883, 907, 911, 947, 977, 991, 1061, 1087, 1093, 1103, 1171, 1213, 1303, 1367, 1373, 1409, 1423

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

This is a member of the family of sequences of primes of the forms x^2 + kxy + y^2.

See for k=1 A007645 = x^2+3y^2, k=2 squares no primes, k=3 A038872, k=4 A068228 = x^2+9y^2, k=5 A139492, k=6 A007519 = x^2+8y^2, k=7 A033212 = x^2+15y^2, k=8 A107152 = x^2+45y^2, k=9 A139493, k=10 A107008 = x^2+24y^2, k=11 A139494, k=12 A139495, k=13 A139496, k=14* = 10 A107008 = x^2+24y^2, k=15 A139497, k=16 A033215 = x^2+21y^2, k=17 A139498, k=18 A107145 = x^2+40y^2, k=19 A139499, k=20 A139500, k=21 A139501, k=22 A139502, k=23 A139503, k=24 A139504, k=25 A139505, k=26,A139506, k=27 A139507, k=28 A139508, k=29 A139509, k=30 A139510, k=31 A139511, k=32 A139512

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 9; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139495 Primes of the form x^2 + 12x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

29, 109, 149, 281, 389, 401, 421, 449, 541, 569, 641, 701, 709, 809, 821, 1009, 1061, 1129, 1201, 1229, 1289, 1381, 1409, 1429, 1481, 1549, 1621, 1709, 1789, 1801, 1901, 2069, 2081, 2129, 2221, 2269, 2381, 2389, 2521, 2549, 2689, 2741, 2801, 2909, 2969

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 12; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)
With[{nn=50},Take[Union[Select[#[[1]]^2+12#[[1]]#[[2]]+#[[2]]^2&/@ Tuples[ Range[ nn],2],PrimeQ]],nn]] (* Harvey P. Dale, Dec 18 2015 *)

A139496 Primes of the form x^2 + 13x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

31, 181, 199, 229, 331, 379, 421, 499, 619, 631, 661, 691, 709, 751, 829, 859, 991, 1021, 1039, 1171, 1279, 1291, 1321, 1489, 1549, 1609, 1621, 1699, 1741, 1831, 1879, 1951, 2011, 2029, 2161, 2179, 2269, 2281, 2311, 2341, 2539, 2671, 2689, 2731, 2971

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 13; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

cp's OEIS Frontend

A228228 Primes congruent to {3, 5, 13, 15} mod 16.

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A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

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A269424 Record (maximal) gaps between primes of the form 8k + 1.

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A269425 Primes 8k + 1 preceding the maximal gaps in A269424.

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A269426 Primes 8k + 1 at the end of the maximal gaps in A269424.

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A022761 n-th 8k+1 prime plus n-th 8k+7 prime.

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A121243 Primes of the form 4*x^2 + 4*x*y + 9*y^2.

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A139493 Primes of the form x^2 + 9x*y + y^2 for x and y nonnegative.

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A121243 Primes of the form 4x^2 + 4xy + 9y^2.