A139487 -id:A139487 - cp's OEIS frontend

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

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

A255233 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

Original entry on oeis.org

5, 7, 13, 9, 21, 11, 17, 29, 19, 15, 31, 37, 17, 27, 33, 23, 29, 21, 41, 47, 37, 23, 43, 33, 49, 55, 51, 31, 41, 69, 53, 29, 43, 59, 35, 31, 45, 61, 41, 67, 85, 57, 47, 63, 43, 53, 35, 75, 93, 37, 71, 61, 83, 47, 89, 39, 73, 53, 63, 79, 49, 85, 69, 97, 103, 109, 55, 65, 47, 77, 67, 83, 49

Offset: 1

Wolfdieter Lang, Feb 18 2015

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007522(n) = -(1 + A139487(n)*8) is given in 2*A255234(n).

For comments and the Nagell reference see A254938.

			The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007522(n) appears as first entry):
  [7, [5, 4]], [23, [7, 6]], [31, [13, 10]],
  [47, [9, 8]], [71, [21, 16]], [79, [11, 10]], [103, [17, 14]], [127, [29, 22]],
  [151, [19, 16]], [167, [15, 14]],
  [191, [31, 24]], [199, [37, 28]],
  [223, [17, 16]], [239, [27, 22]],
  [263, [33, 26]], [271, [23, 20]],
  [311, [29, 24]], [359, [21, 20]],
  [367, [41, 32]], [383, [47, 36]],
  [431, [37, 30]], [439, [23, 22]],
  [463, [43, 34]], [479, [33, 28]], ...
n= 4: 9^2 - 2*(2*4)^2 = -47 = -A007522(4).
a(4) = -(3*5 - 4*(2*3)) = 24 - 15 = 9.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007522 (primes == 7 mod 8), A139487 (8k+7 is prime).

Cf. 2*A255234 (corresponding y2 values), A254938 (x1 values), 2*A255232 (y2 values), A255247, A254936.

apply( {A255233(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3,4]~}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

a(n)^2 - 2*(2*A255234(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A254938(n) - 4*2*A255232(n)), n >= 1.

More terms from Colin Barker, Feb 23 2015

Double-checked and extended by M. F. Hasler, May 22 2025

A023231 Primes p such that 8*p + 7 is also prime.

Original entry on oeis.org

2, 3, 5, 23, 29, 47, 53, 59, 89, 107, 113, 137, 179, 197, 227, 233, 257, 263, 293, 317, 359, 389, 419, 509, 557, 587, 593, 599, 617, 653, 659, 683, 839, 857, 863, 887, 947, 977, 1013, 1097, 1103, 1163, 1193, 1217, 1223, 1229, 1259, 1277, 1283, 1307, 1319, 1409

Offset: 1

David W. Wilson

			For p = 3, 8*p + 7 = 31;
for p = 179, 8*p + 7 = 1439.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A139487, A007522.

[n: n in PrimesUpTo(1500) | IsPrime(8*n+7)]; // Vincenzo Librandi, Nov 20 2010

a := proc (n) if isprime(n) = true and isprime(8*n+7) = true then n else end if end proc: seq(a(n), n = 1 .. 1500); # Emeric Deutsch, Dec 30 2008

Select[Prime@Range@500, PrimeQ[8 # + 7] &] (* Vincenzo Librandi, May 19 2014 *)

Edited by N. J. A. Sloane, Mar 11 2009 at the suggestion of R. J. Mathar

A254766 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

Original entry on oeis.org

5, 11, 9, 17, 13, 23, 21, 17, 27, 35, 23, 21, 41, 31, 29, 39, 37, 53, 33, 31, 41, 59, 39, 49, 37, 35, 43, 63, 53, 37, 49, 77, 59, 47, 75, 83, 65, 53, 73, 51, 45, 61, 71, 59, 79, 69, 95, 55, 49, 101

Offset: 1

Wolfdieter Lang, Feb 11 2015

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = A007522(n) = 7 + 8*A139487(n) is given in A254929(n).
The positive fundamental solutions of the first classes are given in (A254764(n), A254765(n)).
For comments and the Nagell reference see A254764.

			The first pairs [x2(n), y2(n)] of the fundamental positive solutions of the second class are (we list the prime A007522(n) as first entry): [7,[5,3]], [23,[11,7]], [31,[9,5]], [47,[17,11]], [71,[13,7]], [79,[23,15]], [103,[21,13]], [127,[17,9]], [151,[27,17]], [167,[35,23]], [191,[23,13]], [199,[21,11]], [223,[41,27]], [239,[31,19]], [263,[29,17]], [271,[39,25]], ...

Cf. A007522, A139487, A254929, A254764, A254765, A254760, A254761, A254762, A254763,

a(n)^2 - 2*(A254929(n))^2 = A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = 3*A254764(n) - 4*A254765(n), n >= 1.

A023294 Primes that remain prime through 3 iterations of function f(x) = 8x + 7.

Original entry on oeis.org

659, 2549, 5189, 6269, 7229, 7949, 9209, 11579, 16139, 18089, 22739, 25589, 26099, 26339, 29009, 30689, 40289, 51719, 55799, 59669, 60689, 61379, 63599, 69959, 70229, 74609, 85829, 94949, 95819, 102539, 109589, 118169, 121469, 135599, 136889

Offset: 1

David W. Wilson

nonn

Primes p such that 8*p+7, 64*p+63 and 512*p+511 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023231, A023263, and A139487.

[n: n in [1..450000] | IsPrime(n) and IsPrime(8*n+7) and IsPrime(64*n+63) and IsPrime(512*n+511)]; // Vincenzo Librandi, Aug 04 2010

a(n) == 29 (mod 30). - John Cerkan, Sep 23 2016

A023322 Primes that remain prime through 4 iterations of function f(x) = 8x + 7.

Original entry on oeis.org

7949, 25589, 55799, 61379, 69959, 70229, 74609, 174569, 188369, 204719, 220469, 225629, 233759, 250919, 286619, 363659, 552749, 592139, 658349, 735419, 783269, 827549, 931949, 1018889, 1065839, 1126319, 1132739, 1187939, 1215629, 1378529

Offset: 1

David W. Wilson

nonn

Primes p such that 8*p+7, 64*p+63, 512*p+511 and 4096*p+4095 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023231, A023263, A023294, and A139487.

[n: n in [1..5000000] | IsPrime(n) and IsPrime(8*n+7) and IsPrime(64*n+63) and IsPrime(512*n+511) and IsPrime(4096*n+4095)] // Vincenzo Librandi, Aug 04 2010

rp4Q[p_]:=AllTrue[Rest[NestList[8#+7&,p,4]],PrimeQ]; Select[Prime[Range[110000]],rp4Q] (* Harvey P. Dale, Aug 03 2023 *)

a(n) == 9 (mod 10). - John Cerkan, Oct 08 2016

A023350 Primes that remain prime through 5 iterations of function f(x) = 8x + 7.

Original entry on oeis.org

25589, 220469, 225629, 286619, 783269, 1215629, 1407389, 1542029, 1642919, 2329469, 2776979, 3104159, 4082759, 4229129, 5405999, 5905619, 6548849, 6862859, 7681409, 7904669, 8623799, 8971049, 9599309, 9658469, 9725039, 11420579

Offset: 1

David W. Wilson

nonn

Primes p such that 8*p+7, 64*p+63, 512*p+511, 4096*p+4095 and 32768*p+32767 are also primes. - Vincenzo Librandi, Aug 05 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023231, A023263, A023294, A023322, and A139487.

[n: n in [1..19000000] | IsPrime(n) and IsPrime(8*n+7) and IsPrime(64*n+63) and IsPrime(512*n+511) and IsPrime(4096*n+4095) and IsPrime(32768*n+32767)]; // Vincenzo Librandi, Aug 05 2010

i5Q[p_]:=AllTrue[Rest[NestList[8#+7&,p,5]],PrimeQ]; Select[Prime[Range[760000]],i5Q] (* Harvey P. Dale, Jul 05 2025 *)

a(n) == 29 (mod 30). - John Cerkan, Nov 08 2016

A153235 Numbers n such that 8*n+7 is not prime.

Original entry on oeis.org

1, 4, 6, 7, 10, 11, 13, 14, 16, 17, 19, 21, 22, 25, 26, 28, 30, 31, 34, 35, 36, 37, 39, 40, 41, 42, 43, 46, 48, 49, 50, 51, 52, 55, 56, 58, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 79, 81, 82, 83, 84, 85, 86, 87, 88, 91, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Vincenzo Librandi, Dec 21 2008

			Distribution of the terms in the following triangular array:
*;
1,*;
*,*,*;
*,*,7,*;
*,6,*,*,*;
4,*,*,*,17,*;
*,*,*,16,*,*,*;
*,*,14,*,*,*,31,*;
*,11,*,*,*,30,*,*,*;
7,*,*,*,28,*,*,*,49,*:
*,*,*,25,*,*,*,48,*,*,*;
*,*,21,*,*,*,46,*,*,*,71,*; etc.
where * marks the non-integer values of (2*h*k + k + h - 3)/4 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A139487.

[n: n in [0..110] | not IsPrime(8*n+7)]; // Vincenzo Librandi, Jan 12 2013

Select[Range[0, 200], !PrimeQ[8 # + 7] &] (* Vincenzo Librandi, Jan 12 2013 *)

A153236 Numbers n such that 8*n + 3 is not prime.

Original entry on oeis.org

3, 4, 6, 9, 11, 12, 14, 15, 18, 19, 21, 23, 24, 25, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 60, 63, 64, 66, 67, 69, 72, 74, 75, 76, 78, 79, 81, 83, 84, 87, 88, 89, 90, 91, 93, 94, 95, 96, 97

Offset: 1

Vincenzo Librandi, Dec 21 2008

			Distribution of the terms in the following triangular array:
*;
*,*;
*,4,*;
3,*,*,*;
*,*,*,12,*;
*,*,11,*,*,*;
*,9,*,*,*,24,*;
6,*,*,*,23,*,*,*;
*,*,*,21,*,*,*,40,*;
*,*,18,*,*,*,39,*,*,*;
*,14,*,*,*,37,*,*,*,60,*;
9,*,*,*,34,*,*,*,59,*,*,*; etc.
where * marks the non-integer values of (2*h*k + k + h - 1)/4 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007520, A005124, A139487.

[n: n in [0..110] | not IsPrime(8*n+3)]; // Vincenzo Librandi, Jan 12 2013

Select[Range[0, 200], !PrimeQ[8 # + 3] &] (* Vincenzo Librandi, Jan 12 2013 *)

A244087 Numbers n such that 4n+3 and 8n+7 are prime.

Original entry on oeis.org

0, 2, 5, 20, 32, 44, 47, 59, 62, 89, 104, 107, 110, 122, 164, 170, 179, 185, 227, 254, 257, 275, 305, 359, 362, 374, 377, 389, 395, 452, 482, 500, 509, 515, 584, 587, 599, 614, 635, 674, 704, 725, 734, 740, 755, 824, 839, 872, 884, 905, 944, 950, 962, 965, 977

Offset: 1

Vincenzo Librandi, Jun 25 2014

-2 is a primitive root mod (8*n+7).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Intersection of A095278 and A139487.

[n: n in [0..1000] | IsPrime(4*n+3) and IsPrime(8*n+7)];

Select[Range[0, 1500], PrimeQ[4 # + 3]&&PrimeQ[8 # + 7] &]

cp's OEIS Frontend

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

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A255233 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

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A023231 Primes p such that 8*p + 7 is also prime.

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A254766 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

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A023294 Primes that remain prime through 3 iterations of function f(x) = 8x + 7.

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A023322 Primes that remain prime through 4 iterations of function f(x) = 8x + 7.

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A023350 Primes that remain prime through 5 iterations of function f(x) = 8x + 7.

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A244087 Numbers n such that 4n+3 and 8n+7 are prime.