A001132 -id:A001132 - cp's OEIS frontend

A255247 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 {1,7} mod 8).

5, 9, 7, 13, 11, 9, 21, 13, 11, 19, 25, 17, 15, 29, 21, 19, 15, 31, 23, 37, 17, 35, 27, 41, 25, 33, 23, 21, 29, 37, 49, 23, 21, 41, 47, 39, 29, 37, 25, 23, 57, 35, 43, 33, 49, 55, 27, 59, 65, 33, 51, 43, 31, 29, 41, 49, 69, 55, 53, 29, 43, 59, 51, 41, 37, 35

Offset: 1

Wolfdieter Lang, Feb 19 2015

nonn

For the corresponding term y2(n) see A255248(n).
For the positive fundamental proper (sometimes called primitive) solutions x1(n) and y1(n) of the first class of this (generalized) Pell equation see A255235(n) and A255246(n).
The present solutions of this second class are the next to smallest positive ones. Note that for prime 2 only the first class exists.
For the derivation based on the book of Nagell see the comments on A254934 and A254938 for the primes 1 (mod 8) and 7 (mod 8) separately, where also the Nagell reference is given.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A001132(n) is listed as first entry):
  [7, [5, 4]], [17, [9, 7]], [23, [7, 6]],
  [31, [13, 10]], [41, [11, 9]], [47, [9, 8]],
  [71, [21, 16]], [73, [13, 11]], [79, [11, 10]],
  [89, [19, 15]], [97, [25, 19]], [103, [17, 14]],
  [113, [15, 13]], [127, [29, 22]],
  [137, [21, 17]], [151, [19, 16]],
  [167, [15, 14]], [191, [31, 24]],
  [193, [23, 19]], [199, [37, 28]],
  [223, [17, 16]], [233, [35, 27]],
  [239, [27, 22]], [241, [41, 31]], ...
n = 1: 5^2 - 2*4^2 = 25 - 32 = -7 = -A001132(1).
a(3) = -(3*3 - 4*4) = 16 - 9 = 7.

Cf. A001132, A255248, A255235, A255246, A254936, A255233, A254930.

a(n)^2 - 2*A255248(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A255235(n+1) - 4*A255246(n+1)), n >= 1.

More terms from Colin Barker, Feb 26 2015

A095013 Number of 8k+-1 primes (A001132) in range [2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 0, 3, 2, 8, 10, 22, 35, 67, 126, 233, 438, 793, 1525, 2825, 5391, 10192, 19332, 36739, 70163, 133983, 256877, 492962, 946938, 1822776, 3513544, 6780795, 13102754, 25349101, 49090527, 95168113, 184659769, 358635803, 697092152, 1356042601, 2639892053, 5142809798

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A001132, A036378, A095009, A095012, A095014.

a(n) = A036378(n) - A095014(n) = A095009(n) + A095012(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A255248 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 {1,7} mod 8).

Original entry on oeis.org

4, 7, 6, 10, 9, 8, 16, 11, 10, 15, 19, 14, 13, 22, 17, 16, 14, 24, 19, 28, 16, 27, 22, 31, 21, 26, 20, 19, 24, 29, 37, 21, 20, 32, 36, 31, 25, 30, 23, 22, 43, 29, 34, 28, 38, 42, 25, 45, 49, 29, 40, 35, 28, 27, 34, 39, 52, 43, 42, 28, 36, 46, 41, 35, 33, 32

Offset: 1

Wolfdieter Lang, Feb 19 2015

For the corresponding term x2(n) see A255247(n).

See the comments on A255247.

			See A255247.
a(4) = -(2*1 - 3*4) = 12 - 2 = 10.
n=4: 13^2 - 2*10^2 = 169 - 200 = -31 = -A001132(4).

Cf. A001132, A255247, A255235, A255246, A254937, A255234, A254931.

A255247(n)^2 - 2*a(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(2*A255235(n+1) - 3*A255246(n+1)), n >= 1.

More terms from Colin Barker, Feb 26 2015

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).

Original entry on oeis.org

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.

A007519 Primes of form 8n+1, that is, primes congruent to 1 mod 8.

Original entry on oeis.org

17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321, 1361

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Integers n (n > 9) of form 4k + 1 such that binomial(n-1, (n-1)/4) == 1 (mod n) - Benoit Cloitre, Feb 07 2004
Primes of the form x^2 + 8y^2. - T. D. Noe, May 07 2005
Also primes of the form x^2 + 16y^2. See A140633. - T. D. Noe, May 19 2008
Is this the same sequence as A141174?
Being a subset of A001132 and also a subset of A038873, this is also a subset of the primes of the form u^2 - 2v^2. - Tito Piezas III, Dec 28 2008
These primes p are only which possess the property: for every integer m from interval [0, p) with the Hamming distance D(m, p) = 2, there exists an integer h from (m, p) with D(m, h) = 2. - Vladimir Shevelev, Apr 18 2012
Primes p such that p XOR 6 = p + 6. - Brad Clardy, Jul 22 2012
Odd primes p such that -1 is a 4th power mod p. - Eric M. Schmidt, Mar 27 2014
There are infinitely many primes of this form. See Brubaker link. - Alonso del Arte, Jan 12 2017
These primes split in Z[sqrt(2)]. For example, 17 = (-1)(1 - 3*sqrt(2))(1 + 3*sqrt(2)). This is also true of primes of the form 8n - 1. - Alonso del Arte, Jan 26 2017

			a(1) = 17 = 2 * 8 + 1 = (10001)_2. All numbers m from [0, 17) with the Hamming distance D(m, 17) = 2 are 0, 3, 5, 9. For m = 0, we can take h = 3, since 3 is drawn from (0, 17) and D(0, 3) = 2; for m = 3, we can take h = 5, since 5 from (3, 17) and D(3, 5) = 2; for m = 5, we can take h = 6, since 6 from (5, 17) and D(5, 6) = 2; for m = 9, we can take h = 10, since 10 is drawn from (9, 17) and D(9, 10) = 2. - _Vladimir Shevelev_, Apr 18 2012

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Z. I. Borevich and I. R. Shafarevich, Number Theory.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 261.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Ben Brubaker, 18.781, Fall 2007 Problem Set 5: Solutions to Selected Problems, MIT (2007).
Peter Luschny, Binary Quadratic Forms
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Subsequence of A017077 and of A038873.
Cf. A139643. Complement in primes of A154264. Cf. A042987.
Cf. A065091, A002144, A094407, A133870, A142925, A208177, A208178, A076339.
Cf. A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).
Cf. also A242663.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

a007519 n = a007519_list !! (n-1)
a007519_list = filter ((== 1) . a010051) [1,9..]
-- Reinhard Zumkeller, Mar 06 2012

[p: p in PrimesUpTo(2000) | p mod 8 eq 1 ]; // Vincenzo Librandi, Aug 21 2012

Select[1 + 8 Range@ 170, PrimeQ] (* Robert G. Wilson v *)

forprime(p=2,1e4,if(p%8==1,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

forprimestep(p=17,10^4,8, print1(p", ")) \\ Charles R Greathouse IV, Jul 17 2024

lista(nn)= my(vpr = []); for (x = 0, nn, y = 0; while ((v = x^2+6*x*y+y^2) < nn, if (isprime(v), if (! vecsearch(vpr, v), vpr = concat(vpr, v); vpr = vecsort(vpr););); y++;);); vpr; \\ Michel Marcus, Feb 01 2014

A007519_upto(N, start=1)=select(t->t%8==1,primes([start,N]))
#A7519=A007519_upto(10^5)
A007519(n)={while(#A7519A007519_upto(N*3\2, N+1))); A7519[n]} \\ M. F. Hasler, May 22 2025

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 4, -4])
print(Q.represented_positives(1361, 'prime'))  # Peter Luschny, Jan 26 2017

A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

Original entry on oeis.org

7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1151

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Primes that are the sum of no fewer than four positive squares.
Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Primes p such that x^4 = 2 has just two solutions mod p. Subsequence of A040098. Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 = 2 if and only if p - i is a solution mod p of x^4 = 2, so the sum of the two solutions is p. The solutions are given in A065907 and A065908. - Klaus Brockhaus, Nov 28 2001
As this is a subset of A001132, this is also a subset of the primes of form x^2 - 2y^2. And as this is also a subset of A038873, this is also a subset of the primes of form x^2 - 2y^2. - Tito Piezas III, Dec 28 2008
Subsequence of A141164. - Reinhard Zumkeller, Mar 26 2011
Also a subsequence of primes of the form x^2 + y^2 + z^2 + 1. - Arkadiusz Wesolowski, Apr 05 2012
Primes p such that p XOR 6 = p - 6. - Brad Clardy, Jul 22 2012

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Subsequence of A004771.
Cf. A040098, A014754, A065907, A065908, A010051.
Cf. A141174 (d = 32). A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).

a007522 n = a007522_list !! (n-1)
a007522_list = filter ((== 1) . a010051) a004771_list
-- Reinhard Zumkeller, Jan 29 2013

[p: p in PrimesUpTo(2000) | p mod 8 eq 7]; // Vincenzo Librandi, Jun 26 2014

select(isprime, [seq(i,i=7..10000,8)]); # Robert Israel, Nov 22 2016

Select[8Range[200] - 1, PrimeQ] (* Alonso del Arte, Nov 07 2016 *)

(A007522(m) = local(p, s, x, z); forprime(p = 3, m, s = []; for(x = 0, p-1, if(x^4%p == 2%p, s = concat(s, [x]))); z = matsize(s)[2]; if(z == 2, print1(p, ", ")))); A007522(1400)  \\ Does not return a(m) but prints all terms <= m. - Edited to make it executable by M. F. Hasler, May 22 2025.

A007522_upto(N, start=1)=select(p->p%8==7, primes([start, N]))
#A7522=A007522_upto(10^5)
A007522(n)={while(#A7522A007522_upto(N*3\2, N+1))); A7522[n]} \\ M. F. Hasler, May 22 2025

Equals A000040 INTERSECT A004215. - R. J. Mathar, Nov 22 2006

a(n) = 7 + A139487(n)*8, n >= 1. - Wolfdieter Lang, Feb 18 2015

A038873 Primes p such that 2 is a square mod p; or, primes congruent to {1, 2, 7} mod 8.

Original entry on oeis.org

2, 7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599, 601, 607, 617

Offset: 1

N. J. A. Sloane

Same as A001132 except for initial term.
Primes p such that x^2 = 2 has a solution mod p.
The primes of the form x^2 + 2xy - y^2 coincide with this sequence. These are also primes of the form u^2 - 2v^2. - Tito Piezas III, Dec 28 2008
Therefore these are composite in Z[sqrt(2)], as they can be factored as (u^2 - 2v^2)*(u^2 + 2v^2). - Alonso del Arte, Oct 03 2012
After a(1) = 2, these are the primes p such that p^4 == 1 (mod 96). - Gary Detlefs, Jan 22 2014
Also primes of the form 2v^2 - u^2. For example, 23 = 2*4^2 - 3^2. - Jerzy R Borysowicz, Oct 27 2015
Prime factors of A008865 and A028884. - Klaus Purath, Dec 07 2020

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Theorem 5-5, p. 68.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
K. S. Brown, Pythagorean graphs.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index entries for related sequences

Cf. A057126, A087780, A226523, A003629 (complement).
Primes in A035251.
For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

[ p: p in PrimesUpTo(617) | IsSquare(R!2) where R:=ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

seq(`if`(member(ithprime(n) mod 8, {1,2,7}),ithprime(n),NULL),n=1..113); # Nathaniel Johnston, Jun 26 2011

fQ[n_] := MemberQ[{1, 2, 7}, Mod[n, 8]]; Select[ Prime[Range[114]], fQ] (* Robert G. Wilson v, Oct 18 2011 *)

is(n)=isprime(n) && issquare(Mod(2,n)) \\ Charles R Greathouse IV, Apr 23 2015

is(n)=abs(centerlift(Mod(n,8)))<3 && isprime(n) \\ Charles R Greathouse IV, Nov 14 2017

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

A047522 Numbers that are congruent to {1, 7} mod 8.

Original entry on oeis.org

1, 7, 9, 15, 17, 23, 25, 31, 33, 39, 41, 47, 49, 55, 57, 63, 65, 71, 73, 79, 81, 87, 89, 95, 97, 103, 105, 111, 113, 119, 121, 127, 129, 135, 137, 143, 145, 151, 153, 159, 161, 167, 169, 175, 177, 183, 185, 191, 193, 199, 201, 207, 209, 215, 217, 223, 225, 231, 233

Offset: 1

N. J. A. Sloane

Also n such that Kronecker(2,n) = mu(gcd(2,n)). - Jon Perry and T. D. Noe, Jun 13 2003
Also n such that x^2 == 2 (mod n) has a solution. The primes are given in sequence A001132. - T. D. Noe, Jun 13 2003
As indicated in the formula, a(n) is related to the even triangular numbers. - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
Cf. property described by Gary Detlefs in A113801: more generally, these a(n) are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h,n natural numbers). Therefore a(n)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 8). Also a(n)^2 - 1 == 0 (mod 16). - Bruno Berselli, Nov 17 2010
A089911(3*a(n)) = 2. - Reinhard Zumkeller, Jul 05 2013
S(a(n+1)/2, 0) = (1/2)*(S(a(n+1), sqrt(2)) - S(a(n+1) - 2, sqrt(2)))  = T(a(n+1), sqrt(2)/2) = cos(a(n+1)*Pi/4) = sqrt(2)/2 = A010503, identically for n >= 0, where S is the Chebyshev polynomial (A049310) here extended to fractional n, evaluated at x = 0. (For T see A053120.) - Wolfdieter Lang, Jun 04 2023

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001132, A014494, A056575, A010709, A074378, A047336, A056020, A005408, A047209, A007310, A090771, A175885, A091998, A175886, A175887, A058529, A047621, A179260, A352125.
Cf. A077221 (partial sums).
Cf. A000217, A089911, A113801.
Cf. A010503, A049310, A053120.

a047522 n = a047522_list !! (n-1)
a047522_list = 1 : 7 : map (+ 8) a047522_list
-- Reinhard Zumkeller, Jan 07 2012

Select[Range[1, 191, 2], JacobiSymbol[2, # ]==1&]

a(n)=4*n-2+(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = sqrt(8*A014494(n)+1) = sqrt(16*ceiling(n/2)*(2*n+1)+1) = sqrt(8*A056575(n)-8*(2n+1)*(-1)^n+1). - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
1 - 1/7 + 1/9 - 1/15 + 1/17 - ... = (Pi/8)*(1 + sqrt(2)). [Jolley] - Gary W. Adamson, Dec 16 2006
From R. J. Mathar, Feb 19 2009: (Start)
a(n) = 4n - 2 + (-1)^n = a(n-2) + 8.
G.f.: x(1+6x+x^2)/((1+x)(1-x)^2). (End)
a(n) = 8*n - a(n-1) - 8. - Vincenzo Librandi, Aug 06 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 8*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
E.g.f.: 1 + (4*x - 1)*cosh(x) + (4*x - 3)*sinh(x). - Stefano Spezia, May 13 2021
E.g.f.: 1 + (4*x - 3)*exp(x) + 2*cosh(x). - David Lovler, Jul 16 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2+sqrt(2)) (A179260).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/8)*cosec(Pi/8) (A352125). (End)

A058529 Numbers whose prime factors are all congruent to +1 or -1 modulo 8.

Original entry on oeis.org

1, 7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 127, 137, 151, 161, 167, 191, 193, 199, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 289, 311, 313, 329, 337, 343, 353, 359, 367, 383, 391, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487

Offset: 1

William Bagby (bagsbee(AT)aol.com), Dec 24 2000

Numbers of the form x^2 - 2*y^2, where x is odd and x and y are relatively prime. - Franklin T. Adams-Watters, Jun 24 2011
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1, a <= b); sequence gives values b-a, sorted with duplicates removed; terms > 1 in sequence give values of a + b, sorted. (See A046086 and A046087.)
Ordered set of (semiperimeter + radius of largest inscribed circle) of all primitive Pythagorean triangles. Semiperimeter of Pythagorean triangle + radius of largest circle inscribed in triangle = ((a+b+c)/2) + ((a+b-c)/2) = a + b.
The terms of this sequence are all of the form 6*N +- 1, since the prime divisors are, and numbers of this form are closed under multiplication. In fact, all terms are == 1, 7, 17, or 23 (mod 24). - J. T. Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 28 2009, edited by Franklin T. Adams-Watters, Jun 24 2011
Is similar to A001132, but includes composites whose factors are in A001132. Can be generated in this manner.
Third side of primitive parallepipeds with square base; that is, integer solution of a^2 + b^2 + c^2 = d^2 with gcd(a,b,c) = 1 and b = c. - Carmine Suriano, May 03 2013
Other than -1, values of difference z-y that solve the Diophantine equation x^2 + y^2 = z^2 + 2. - Carmine Suriano, Jan 05 2015
For k > 1, k is in the sequence iff A330174(k) > 0. - Ray Chandler, Feb 26 2020

B Berggren, Pytagoreiska trianglar. Tidskrift för elementär matematik, fysik och kemi, 17:129-139, 1934.
Olaf Delgado-Friedrichs and Michael O’Keeffe, Edge-transitive lattice nets, Acta Cryst. (2009). A65, 360-363.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
F. Barnes, primitive Pythagorean triangles where a-b is a constant.
Johannes Boot, Draft English translation of B Berggren's (1934) article "Pytagoreiska Trianglar", ResearchGate 2017.
K. S. Brown, Pythagorean graphs.
O. Delgado-Friedrichs and M. O'Keeffe, Edge-transitive lattice nets, Acta Cryst. A, A65 (2009), 360-363.
B. Frénicle, Méthode pour trouver la solution des problèmes par les exclusions, 44 pages (see p. 31). In Divers ouvrages de mathematique .. Par Messieurs de l'Academie Royale des Sciences, in-fol, 6+518+1PP, Paris, 1693. - _Paul Curtz_, Sep 06 2008

Cf. A020882-A020886, A020888, A046086, A046087, A014498, A001132, A001653, A027748, A047522, A330174.

a058529 n = a058529_list !! (n-1)
a058529_list = filter (\x -> all (`elem` (takeWhile (<= x) a001132_list))
                                 $ a027748_row x) [1..]
-- Reinhard Zumkeller, Jan 29 2013

Select[Range[500], Union[Abs[Mod[Transpose[FactorInteger[#]][[1]], 8, -1]]] == {1} &] (* T. D. Noe, Feb 07 2012 *)

is(n)=my(f=factor(n)[,1]%8); for(i=1,#f, if(f[i]!=1 && f[i]!=7, return(0))); 1 \\ Charles R Greathouse IV, Aug 01 2016

a(n) = |A-B|=|j^2-2*k^2|, j=(2*n-1), k,n in N, GCD(j,k)=1, the absolute difference between primitive Pythagorean triple legs (sides adjacent to the right angle). - Roger M Ellingson, Dec 09 2023

More terms from Naohiro Nomoto, Jul 02 2001
Edited by Franklin T. Adams-Watters, Jun 24 2011
Duplicated comment removed and name rewritten by Wolfdieter Lang, Feb 17 2015

cp's OEIS Frontend

A255247 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 {1,7} mod 8).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A095013 Number of 8k+-1 primes (A001132) in range [2^n,2^(n+1)].

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A255248 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 {1,7} mod 8).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A007519 Primes of form 8n+1, that is, primes congruent to 1 mod 8.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

PARI

PARI

SageMath

A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI