A007519 -id:A007519 - cp's OEIS frontend

A254930 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1 (primes congruent to 1 or 7 mod 8).

5, 7, 11, 9, 13, 17, 13, 19, 23, 17, 15, 21, 25, 17, 23, 27, 35, 23, 29, 21, 41, 25, 31, 23, 35, 29, 39, 43, 37, 31, 27, 49, 53, 33, 31, 37, 47, 41, 55, 59, 31, 45, 39, 49, 37, 35, 61, 37, 35

Offset: 1

Wolfdieter Lang, Feb 12 2015

The corresponding terms y = y2(n) are given in A254931(n).
There is only one fundamental solution for prime 2 (no second class exists), and this solution (x, y) has been included in (A002334(1), A002335(1)) = (2, 1).
The second class x sequence for the primes 1 (mod 8), which are given in A007519, is A254762, and for the primes 7 (mod 8), given in A007522, it is A254766.
The second class solutions give the second smallest positive integer solutions of this Pell equation.
For comments and the Nagell reference see A254760.

			n = 3: 11^2 - 2*7^2 = 23 = A001132(3) = A007522(2).
The first pairs of these second class solutions [x2(n), y2(n)] are (a star indicates primes congruent to 1 (mod 8)):
n  A001132(n)   a(n)  A254931(n)
1     7           5        3
2    17 *         7        4
3    23          11        7
4    31           9        5
5    41 *        13        8
6    47          17       11
7    71          13        7
8    73 *        19       12
9    89 *        17       10
10   97 *        15        8
11  103          21       13
12  113 *        25       16
13  127          17        9
14  137 *        23       14
15  151          27       17
16  167          35       23
17  191          23       13
18  193 *        29       18
19  199          21       11
20  223          41       27
...

Cf. A001132, A254931, A002334, A002335, A007519, A254762, A007522, A254766, A254760.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; x2 = xy[[-1, 1]] // Simplify; Print[x2]; Sow[x2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

a(n)^2 - 2*A254931(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer solving this (generalized) Pell equation.

a(n) = 3*A002334(n+1) - 4*A002335(n+1), n >= 1.

A254931 Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1, (primes congruent to 1 or 7 mod 8).

Original entry on oeis.org

3, 4, 7, 5, 8, 11, 7, 12, 15, 10, 8, 13, 16, 9, 14, 17, 23, 13, 18, 11, 27, 14, 19, 12, 22, 17, 25, 28, 23, 18, 14, 32, 35, 19, 17, 22, 30, 25, 36, 39, 16, 28, 23, 31, 21, 19, 40, 20, 18, 38

Offset: 1

Wolfdieter Lang, Feb 12 2015

The corresponding terms x = x2(n) are given in A254930(n).
The y2-sequence for the second class for the primes congruent to 1 (mod 8), which are given in A007519, is 2*A254763. For the primes congruent to 7 (mod 8), given in A007522, the y2-sequence is A254929.
For comments and the Nagell reference see A254760.

			a(4) = 2*7 - 3*3 = 5.
A254930(4)^2 - 2*a(4)^2 = 9^2 - 2*5^2 = 31 = A001132(4) = A007522(3).
See A254930 for the first pairs (x2(n), y2(n)).

Cf. A254930, A001132, A002334, A002335, A007519, 2*A254763, A007522, A254929, A254760.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; y2 = xy[[-1, 2]] // Simplify; Print[y2]; Sow[y2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

A254930(n)^2 - 2*a(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer satisfying this (generalized) Pell equation.

a(n) = 2*A002334(n+1) - 3*A002335(n+1), n >= 1.

A258149 Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.

Original entry on oeis.org

1, 0, 7, 7, 0, 17, 0, 1, 0, 31, 23, 0, 0, 0, 49, 0, 17, 0, 23, 0, 71, 47, 0, 7, 0, 41, 0, 97, 0, 41, 0, 7, 0, 0, 0, 127, 79, 0, 31, 0, 0, 0, 89, 0, 161, 0, 73, 0, 17, 0, 47, 0, 119, 0, 199, 119, 0, 0, 0, 1, 0, 73, 0, 0, 0, 241

Offset: 2

Wolfdieter Lang, Jun 10 2015

For primitive Pythagorean triangles characterized by certain (n,m) pairs and references see A225949.
Here a(n,m) = 0 for non-primitive Pythagorean triangles, and for primitive Pythagorean triangles a(n,m) = abs(n^2 - m^2 - 2*n*m) = abs((n-m)^2 - 2*m^2).
The number of non-vanishing entries in row n is A055034(n).
D(n,m):= n^2 - m^2 - 2*n*m >= 0 if 1 <= m <= floor(n/(sqrt(2)+1)), and D(n,m) < 0 if n/(sqrt(2)+1)+1 <= m <= n-1, for n >= 2.
The Pell equation (n-m)^2 - 2*m^2 = +/- N is important here to find the representations of +N or -N in the triangle D(n,m). For instance, odd primes N have to be of the +1 (mod 8) (A007519) or -1 (mod 8) (A007522) form, that is, from A001132. See the Nagell reference, Theorem 110, p. 208 with Theorem 111, pp. 210-211. E.g., N = +7 appears for m = 1, 3, 9, 19, 53, ... (A077442) for n = 4, 8, 22, 46, 128, ... (2*A006452).
  N = -7 appears for n = 3, 9, 19, 53, 111, ... (A077442) and m = 2, 4, 8, 22, 46, ... (2*A006452).
For the  signed version 2*n*m - (n^2 - m^2) see A278717. - Wolfdieter Lang, Nov 30 2016

			The triangle a(n,m) begins:
n\m   1  2  3  4  5  6  7   8   9  10  11 ...
2:    1
3:    0  7
4:    7  0 17
5:    0  1  0 31
6:   23  0  0  0 49
7:    0 17  0 23  0 71
8:   47  0  7  0 41  0 97
9:    0 41  0  7  0  0  0 127
10:  79  0 31  0  0  0 89   0 161
11:   0 73  0 17  0 47  0 119   0 199
12: 119  0  0  0  1  0 73   0   0   0 241
...
a(2,1) = |1^2 - 2*1^2| = 1 for the primitive Pythagorean triangle (pPt) [3,4,5] with |3-4| = 1.
a(3,2) = |1^2 - 2*2^2| = 7 for the pPt [5,12,13] with |5 - 12| = 7.
a(4,1) = |3^2 - 2*1^2| = 7 for the pPt [15, 8, 17] with |15 - 8| = 7.

See also A225949.
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, pp. 208, 210-211.

Cf. A249866, A222946, A225949, A222951, A258150, A278717 (signed).

a[n_, m_] /; n > m >= 1 && CoprimeQ[n, m] && (-1)^(n+m) == -1 := Abs[n^2 - m^2 - 2*n*m]; a[, ] = 0; Table[a[n, m], {n, 2, 12}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Jun 16 2015, after given formula *)

a(n,m) = abs(n^2 - m^2 -2*n*m) = abs((n-m)^2 - 2*m^2) if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^(n+m) = -1); otherwise a(n,m) = 0.

A269704 Numbers k such that prime(k) == 1 (mod 8).

Original entry on oeis.org

7, 13, 21, 24, 25, 30, 33, 44, 51, 53, 55, 60, 65, 68, 71, 79, 80, 84, 87, 88, 98, 104, 106, 108, 110, 113, 116, 122, 135, 136, 140, 148, 152, 158, 159, 162, 165, 169, 174, 176, 184, 189, 191, 196, 197, 199, 204, 209, 211, 216, 218, 223, 227, 234, 237, 245

Offset: 1

Vincenzo Librandi, Mar 04 2016

nonn

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

			a(1) = 7 because prime(7) = 17 and 17 == 1 (mod 8).

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

The associated primes are in A007519.

Cf. similar sequences listed in A269703.

[n: n in [1..500] | NthPrime(n) mod 8 eq 1];

Res:= NULL: count:= 0:
p:= 2:
for n from 2 while count < 100 do
  p:= nextprime(p);
  if p mod 8 = 1 then count:= count+1; Res:= Res, n fi
od:
Res; # Robert Israel, May 06 2019

Select[Range[300], Mod[Prime[#], 8] == 1 &]

lista(nn) = for(n=1, nn, if(Mod(prime(n),8)==1, print1(n, ", "))); \\ Altug Alkan, Mar 04 2016

A337145 a(n) is the determinant of the 2 X 2 matrix whose entries (when read by rows) are the n-th primes congruent to 1, 3, 5, 7 mod 8 respectively.

Original entry on oeis.org

104, 800, 1712, 2592, 3760, 4840, 5728, 12848, 15664, 18424, 20888, 23520, 28232, 28560, 25320, 30280, 37248, 50520, 43680, 33664, 61560, 73920, 70544, 57696, 38696, 27408, 79280, 63392, 107328, 109536, 162608, 172296, 187352, 197040, 248064, 228320, 215912, 229152, 255480, 231304, 286408, 256320

Offset: 1

J. M. Bergot and Robert Israel, Jan 27 2021

The first negative term is a(20750) = -58207896.

All terms are divisible by 8.

			The first primes == 1, 3, 5, 7 (mod 8) are 17, 3, 5, 7 respectively, so a(1) = 17*7 - 3*5 = 104.
The second primes == 1, 3, 5, 7 (mod 8) are 41, 11, 13, 23 respectively, so a(2) = 41*23 - 11*13 = 800.
The third primes == 1, 3, 5, 7 (mod 8) are 73, 19, 29, 31 respectively, so a(3) = 73*31 - 19*29 = 1712.

Robert Israel, Table of n, a(n) for n = 1..30000

Cf. A335581, A335592, A337146, A337147.

Cf. A007519, A007520, A007521, A007522.

R:= NULL:
L:= [-7, -5, -3, -1]:
found:= false:
for k from 1 to 100 do
  for i from 1 to 4 do
   for x from L[i]+8 by 8 do until isprime(x);
   L[i]:= x;
  od;
  v:= L[1]*L[4]-L[2]*L[3];
  R:= R,v;
od:
R;

A020671 Numbers of form x^2 + 8 y^2.

Original entry on oeis.org

0, 1, 4, 8, 9, 12, 16, 17, 24, 25, 32, 33, 36, 41, 44, 48, 49, 57, 64, 68, 72, 73, 76, 81, 88, 89, 96, 97, 100, 108, 113, 121, 128, 129, 132, 136, 137, 144, 152, 153, 164, 169, 172, 176, 177, 192, 193, 196, 200, 201, 204, 209, 216, 225, 228, 233, 236, 241, 249, 256, 257

Offset: 1

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A007519.

[n: n in [0..230] | NormEquation(8, n) eq true]; // Vincenzo Librandi, Aug 31 2016

lim=260; k=8; Union@Flatten@Table[x^2 + k y^2, {y, 0, Sqrt[lim/k]}, {x, 0, Sqrt[lim-k y^2]}] (* Vincenzo Librandi, Aug 31 2016 *)

A125039 Primes of the form 8k+1 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^4 + 1}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

17, 1336337, 4261668267710686591310687815697, 41, 4390937134822286389262585915435960722186022220433, 241, 1553, 243537789182873, 97, 27673, 4289, 457, 137201, 73, 337, 569891669978849, 617, 1697, 65089, 1609, 761

Offset: 1

Nick Hobson, Nov 18 2006

nonn

All prime divisors of (2Q)^4 + 1 are congruent to 1 modulo 8.

			a(3) = 4261668267710686591310687815697 is the smallest prime divisor of (2Q)^4 + 1 = 4261668267710686591310687815697, where Q = 17 * 1336337.

G. A. Jones and J. M. Jones, Elementary Number Theory, Springer-Verlag, NY, (1998), p. 271.

Sean A. Irvine, Table of n, a(n) for n = 1..29

Cf. A000945, A007519, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

More terms from Sean A. Irvine, Apr 09 2015

A139505 Primes of the form x^2 + 25x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

151, 163, 307, 397, 409, 541, 547, 601, 673, 811, 823, 859, 967, 997, 1153, 1231, 1237, 1327, 1567, 1669, 1741, 1879, 2083, 2143, 2281, 2293, 2557, 2677, 2707, 2833, 2971, 3037, 3259, 3313, 3433, 3877, 4003, 4129, 4153, 4603, 4639, 4861, 4957, 5101, 5227

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..5000
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 = 25; 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=80},Select[Union[#[[1]]^2+25#[[1]]#[[2]]+#[[2]]^2&/@Tuples[ Range[ 0,nn],2]],PrimeQ[#]&&#Harvey P. Dale, Feb 10 2020 *)

A210630 Decimal expansion of Product_{primes p == 1 (mod 8)} p*(p-8)/(p-4)^2.

Original entry on oeis.org

8, 8, 3, 0, 7, 1, 0, 0, 4, 7, 4, 3, 9, 4, 6, 6, 7, 1, 4, 1, 7, 8, 3, 4, 2, 9, 9, 0, 0, 3, 1, 0, 8, 5, 3, 4, 6, 7, 6, 8, 8, 8, 8, 3, 4, 8, 8, 0, 9, 7, 3, 4, 7, 0, 7, 1, 9, 2, 9, 5, 1, 5, 9, 3, 9, 5, 2, 1, 1, 9, 4, 6, 9, 9, 0, 6, 5, 6, 5, 9, 6, 8, 8, 5, 7, 9, 9, 3, 8, 3, 2, 8, 6, 0, 3, 7, 9, 1, 6, 4, 6, 3, 5, 8, 5, 2

Offset: 0

R. J. Mathar, Mar 25 2012

Equals the product_{s>=2} of 1/zeta_(8,1)(s)^gamma(s), where gamma(s) = 16, 128, 888, 6144, 42256, 293912,... is an Euler transformation of the associated polynomial (1/x)(1/x-8)/(1/x-4)^2, and where the zeta_(m,n)(s) are the zeta prime modulo functions defined in section 3.3 of arXiv:1008.2547.

Note that Product_{k>=1} (8*k-7) * (8*k+1) / (8*k-3)^2 = Pi * 2^(9/2) * Gamma(1/4)^2 / Gamma(1/8)^4 = 0.290040073098462288674... - Vaclav Kotesovec, May 13 2020

			0.88307100474394667141783429900310853467688883488097347...

Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.9).
Daniel Shanks, Lal's constant and generalizations, Math. Comp. 21 (100) (1967) 705-707.

Cf. A007519, A334826.

$MaxExtraPrecision = 1000; digits = 121;
f[p_] := p*(p - 8)/(p - 4)^2;
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*(P[8, 1, m] - 1/17^m); sump = sump + difp; m++];
RealDigits[Chop[N[f[17]*Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020

More digits from Vaclav Kotesovec, Jan 16 2021

A228227 Primes congruent to {7, 11} mod 16.

Original entry on oeis.org

7, 11, 23, 43, 59, 71, 103, 107, 139, 151, 167, 199, 251, 263, 283, 311, 331, 347, 359, 379, 439, 443, 487, 491, 503, 523, 571, 587, 599, 619, 631, 647, 683, 727, 743, 811, 823, 827, 839, 859, 887, 907, 919, 967, 971, 983, 1019, 1031, 1051, 1063, 1163, 1223

Offset: 1

Arkadiusz Wesolowski, Aug 16 2013

nonn

Union of A141194 and A141195.

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. Therefore A060953(a(n)) = 0.

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, A141194, A141195, A228228.

[p: p in PrimesUpTo(1223) | p mod 16 in {7, 11}];

Select[Prime@Range[200], MemberQ[{7, 11}, Mod[#, 16]] &]

cp's OEIS Frontend

A254930 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1 (primes congruent to 1 or 7 mod 8).

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A254931 Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1, (primes congruent to 1 or 7 mod 8).

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A258149 Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.

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A269704 Numbers k such that prime(k) == 1 (mod 8).

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A337145 a(n) is the determinant of the 2 X 2 matrix whose entries (when read by rows) are the n-th primes congruent to 1, 3, 5, 7 mod 8 respectively.

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A020671 Numbers of form x^2 + 8 y^2.

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A125039 Primes of the form 8k+1 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^4 + 1}, where Q is the product of previous terms in the sequence.

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A139505 Primes of the form x^2 + 25x*y + y^2 for x and y nonnegative.

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