A038873 -id:A038873 - cp's OEIS frontend

A028886 Primes of the form k^2 - 8.

17, 41, 73, 113, 281, 353, 433, 521, 617, 953, 1217, 1361, 2017, 2393, 2593, 2801, 4217, 4481, 6553, 7561, 8273, 8641, 10193, 10601, 13217, 13681, 14153, 14633, 15121, 16633, 17681, 18217, 20441, 21017, 21601, 22193, 25913, 26561, 29921

Offset: 1

Patrick De Geest

Primes of A028884. Also subsequence of A038873. - Klaus Purath, Jan 28 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..8000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x)

Cf. A028884, A038873.

[a: n in [3..300] | IsPrime(a) where a is n^2-8]; // Vincenzo Librandi, Dec 01 2011

Select[Range[3,1000]^2-8,PrimeQ] (* Vincenzo Librandi, Dec 01 2011 *)

A059770 First solution of x^2 = 2 mod p for primes p such that a solution exists.

Original entry on oeis.org

0, 3, 6, 5, 8, 17, 7, 12, 32, 9, 25, 14, 38, 51, 16, 31, 46, 13, 57, 52, 20, 15, 85, 99, 22, 60, 110, 96, 132, 66, 120, 26, 167, 19, 79, 137, 53, 97, 188, 206, 21, 30, 80, 203, 187, 91, 157, 249, 201, 34, 142, 166, 222, 194, 296, 94, 67, 36, 283, 324, 27, 102, 113, 73

Offset: 1

Klaus Brockhaus, Feb 21 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. For p > 2: If x^2 = 2 has a solution mod p, then it has exactly two solutions and their sum is p; i is a solution mod p of x^2 = 2 iff p-i is a solution mod p of x^2 = 2. No integer occurs more than once in this sequence. Moreover, no integer (except 0) occurs both in this sequence and in sequence A059771 of the second solutions (Cf. A059772).

			a(6) = 17, since 41 is the sixth term of A038873, 17 and 24 are the solutions mod 41 of x^2 = 2 and 17 is the smaller one.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
K. Matthews, Finding square roots mod p by Tonelli's algorithm
R. Chapman, Square roots modulo a prime

Cf. A038873, A059771, A059772.

fQ[n_] := MemberQ[{1, 2, 7}, Mod[n, 8]]; f[n_] := PowerMod[2, 1/2, n]; f@ Select[ Prime[Range[135]], fQ] (* Robert G. Wilson v, Oct 18 2011 *)

a(n) = first (least) solution of x^2 = 2 mod p, where p is the n-th prime such that x^2 = 2 mod p has a solution, i.e. p is the n-th term of A038873.

A059771 Second solution of x^2 = 2 mod p for primes p such that a solution exists.

Original entry on oeis.org

0, 4, 11, 18, 23, 24, 40, 59, 41, 70, 64, 83, 65, 62, 111, 106, 105, 154, 134, 141, 179, 208, 148, 140, 219, 197, 153, 175, 149, 245, 193, 311, 186, 340, 288, 246, 348, 312, 243, 227, 418, 419, 377, 260, 292, 396, 346, 272, 368, 543, 451, 433, 379, 413, 321

Offset: 1

Klaus Brockhaus, Feb 21 2001

Solutions mod p are represented by integers from 0 to p-1. For p > 2: If x^2 = 2 has a solution mod p, then it has exactly two solutions and their sum is p; i is a solution mod p of x^2 = 2 iff p-i is a solution mod p of x^2 = 2. No integer occurs more than once in this sequence. Moreover, no integer (except 0) occurs both in this sequence and in sequence A059770 of the first solutions (Cf. A059772).

			a(6) = 24 since 41 is the sixth term of A038873, 17 and 24 are the solutions mod 41 of x^2 = 2 and 24 is the larger one.

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

Cf. A038873, A059770, A059772.

R:= 0: p:= 2: count:= 1:
while count < 100 do
  p:= nextprime(p);
  if NumberTheory:-QuadraticResidue(2,p)=1 then
    v:= NumberTheory:-ModularSquareRoot(2,p);
    R:= R, max(v,p-v);
    count:= count+1
  fi
od:
R; # Robert Israel, Sep 07 2023

a(n) = second (larger) solution of x^2 = 2 mod p, where p is the n-th prime such that x^2 = 2 mod p has a solution, i.e. p is the n-th term of A038873. a(n) = 0 if x^2 = 2 mod p has one solution (only for p = 2).

A060515 Integers i > 1 for which there is no prime p such that i is a solution mod p of x^2 = 2.

Original entry on oeis.org

2, 10, 28, 39, 45, 54, 58, 74, 87, 88, 101, 108, 114, 116, 130, 143, 147, 156, 164, 168, 178, 180, 181, 225, 228, 235, 238, 242, 244, 248, 256, 263, 270, 271, 277, 304, 305, 317, 318, 325, 333, 334, 338, 347, 363, 367, 373, 374, 378, 380, 381, 386, 397, 402

Offset: 1

Klaus Brockhaus, Mar 24 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^2 = 2 iff i^2-2 has a prime factor > i; i is a solution mod p of x^2 = 2 iff p is a prime factor of i^2-2 and p > i.

			a(1) = 2, since there is no prime p such that 2 is a solution mod p of x^2 = 2. a(2) = 10, since there is no prime p such that 10 is a solution mod p of x^2 = 2 and for each integer i from 3 to 9 there is a prime q such that i is a solution mod q of x^2 = 2 (cf. A059772).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A038873, A059772.

select(t -> max(numtheory:-factorset(t^2-2)) <= t, [$2..1000]); # Robert Israel, Feb 23 2016

is(n)=my(f=factor(n^2-2)[,1]);n>1&&f[#f]<=n \\ Charles R Greathouse IV, Aug 24 2013

Integer i > 1 is a term of this sequence iff i^2-2 has no prime factor > i.

A087780 Number of non-congruent solutions to x^2 == 2 mod n.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A038873, A057126, A060594.

Cf. A013661, A196525.

f[2, e_] := Boole[e == 1]; f[p_, e_] := If[MemberQ[{1, 7}, Mod[p, 8]], 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, (f[i,2] == 1), if(f[i,1]%8 == 1 || f[i,1]%8 == 7, 2, 0)));} \\ Amiram Eldar, Nov 21 2023

def A087780(n) :
    res = 1
    for (p, m) in factor(n) :
        if p % 8 in [1, 7] : res *= 2
        elif not (p==2 and m==1) : return 0
    return res
# Eric M. Schmidt, Apr 20 2013

Multiplicative with a(p^m) = 2 for p == 1, 7 (mod 8); a(p^m) = 0 for p == 3, 5 (mod 8); a(2^1) = 1; a(2^m) = 0 for m > 1. - Eric M. Schmidt, Apr 20 2013

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(sqrt(2)+1)/(sqrt(2)*zeta(2)) = A196525/A013661 = 0.37887551404073012021... . - Amiram Eldar, Nov 21 2023

More terms from David Wasserman, Jun 17 2005

A097958 Primes p such that p divides 6^((p-1)/2) - 3^((p-1)/2).

Original entry on oeis.org

3, 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

Cino Hilliard, Sep 06 2004

Apart from the first term, the same as A001132 or A038873. - Jianing Song, Apr 21 2022

Jianing Song, Table of n, a(n) for n = 1..10000 (terms 1..998 from Harvey P. Dale)

Cf. A001132, A038873.

Select[Prime[Range[150]],Divisible[6^((#-1)/2)-3^((#-1)/2),#]&] (* Harvey P. Dale, Dec 25 2021 *)

\s = +-1,d=diff ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }

isA097958(p) = (p==3) || (isprime(p) && kronecker(p,2)==1) \\ Jianing Song, Apr 21 2022

Equals {3} union A001132. - Jianing Song, Apr 21 2022

Definition corrected by Cino Hilliard, Nov 10 2008
Definition clarified by Harvey P. Dale, Dec 25 2021
Offset corrected by Jianing Song, Apr 21 2022

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).

Original entry on oeis.org

2, 3, 19, 23, 37, 41, 61, 67, 71, 73, 79, 89, 97, 109, 127, 137, 149, 173, 181, 211, 223, 227, 251, 257, 269, 283, 293, 311, 317, 347, 349, 353, 359, 367, 373, 383, 389, 397, 401, 419, 439, 457, 461, 463, 479, 487, 499, 503, 509, 523, 547, 557, 587, 593, 607

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 03 2008

nonn

Discriminant = 73. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac.

Is this the same as A038957? - R. J. Mathar, Jul 04 2008. Answer: almost certainly - see the Tunnell notes in A033212. - N. J. A. Sloane, Oct 18 2014

			a(2) = 3 because we can write 3 = 4*1^2 + 3*1*1 - 4*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

Original entry on oeis.org

3, 5, 7, 17, 23, 37, 73, 97, 107, 113, 163, 167, 173, 193, 197, 227, 233, 277, 283, 313, 317, 337, 347, 367, 397, 487, 503, 547, 607, 617, 643, 653, 673, 677, 683, 743, 787, 823, 827, 853, 857, 877, 887, 907, 947, 983, 997, 1013, 1093, 1117, 1153, 1163, 1187

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008

nonn

Discriminant = 85. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

			a(1) = 3 because we can write 3 = 3*1^2 + 5*1*0 - 5*0^2 (or 3 = 7*0^2 + 13*0*1 + 3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A141773 (d=85). See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141750 (d=73). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

More terms from Colin Barker, Apr 04 2015

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).

Original entry on oeis.org

2, 5, 11, 17, 47, 53, 67, 71, 73, 79, 89, 97, 107, 109, 131, 139, 157, 167, 173, 179, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 283, 307, 311, 317, 331, 347, 367, 373, 401, 409, 443, 449, 461, 463, 467, 479, 487, 509, 523, 587, 601, 607, 613, 619, 631

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008

nonn

Discriminant = 89. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

A subsequence of (and may possibly coincide with) A038977. - R. J. Mathar, Jul 22 2008

			a(1) = 2 because we can write 2 = 4*1^2 + 3*1*1 - 5*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141750 (d=73). A141772, A141773 (d=85). A141776, A141777 (d=88). A141778 (d=89). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A142955 Primes of the form 3x^2 + 4xy - 5y^2 (as well as of the form 3x^2 + 10xy + 2y^2).

Original entry on oeis.org

2, 3, 19, 31, 59, 67, 71, 79, 103, 107, 127, 151, 167, 179, 211, 223, 227, 307, 331, 379, 383, 431, 439, 487, 523, 547, 563, 599, 607, 659, 683, 743, 751, 787, 811, 827, 839, 863, 887, 907, 911, 971, 983, 991, 1019, 1039, 1063, 1091, 1123, 1171, 1231, 1283

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (laucabfer(AT)alum.us.es), Jul 14 2008

nonn

Discriminant = 76. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

			a(4) = 31 because we can write 31 = 3*3^2 + 4*3*2 - 5*2^2 (or 31 = 3*1^2 + 10*1*2 + 2*2^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A142956 (d=76). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

More terms from Colin Barker, Apr 05 2015

Edited by M. F. Hasler, Feb 18 2022

cp's OEIS Frontend

A028886 Primes of the form k^2 - 8.

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A059770 First solution of x^2 = 2 mod p for primes p such that a solution exists.

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A059771 Second solution of x^2 = 2 mod p for primes p such that a solution exists.

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A060515 Integers i > 1 for which there is no prime p such that i is a solution mod p of x^2 = 2.

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Maple

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A087780 Number of non-congruent solutions to x^2 == 2 mod n.

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Sage

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A097958 Primes p such that p divides 6^((p-1)/2) - 3^((p-1)/2).

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Mathematica

PARI

PARI

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A141750 Primes of the form 4*x^2 + 3*x*y - 4*y^2 (as well as of the form 2*x^2 + 9*x*y + y^2).

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A141772 Primes of the form 3*x^2 + 5*x*y - 5*y^2 (as well as of the form 7*x^2 + 13*x*y + 3*y^2).

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).

A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).

A142955 Primes of the form 3x^2 + 4xy - 5y^2 (as well as of the form 3x^2 + 10xy + 2y^2).