A141174 -id:A141174

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A007519

Primes of form 8n+1, that is, primes congruent to 1 mod 8.
(Formerly M5037)

+10
99

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

REFERENCES

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).

LINKS

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

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.

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

(Haskell)

a007519 n = a007519_list !! (n-1)

a007519_list = filter ((== 1) . a010051) [1, 9..]

-- Reinhard Zumkeller, Mar 06 2012

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

(PARI) 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

(SageMath) # 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

CROSSREFS

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.

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

A007522

Primes of the form 8n+7, that is, primes congruent to -1 mod 8.
(Formerly M4376)

+10
82

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

REFERENCES

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.

LINKS

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)

FORMULA

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

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

MAPLE

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

MATHEMATICA

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

PROG

(PARI) 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)

(Haskell)

a007522 n = a007522_list !! (n-1)

a007522_list = filter ((== 1) . a010051) a004771_list

-- Reinhard Zumkeller, Jan 29 2013

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

CROSSREFS

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).

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

A141373

Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2).

+10
9

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - T. D. Noe, May 19 2008

REFERENCES

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

LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]

William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy] See Table II.

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.

FORMULA

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

EXAMPLE

19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

MATHEMATICA

QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)

PROG

(Magma) [3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

(PARI) list(lim)=my(v=List(), w, t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\16), if(isprime(t=w+16*y^2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

CROSSREFS

Cf. A107132, A139827, A107003, A141375, A141376 (d=96).

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),

A141336, A141337 (d=92),

A141338, A141339 (d=93),

A141161, A141163 (d=148),

A141165, A141166 (d=229),

A141167, A141167, A141167 (d=257).

KEYWORD

nonn

AUTHOR

T. D. Noe, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

EXTENSIONS

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.

STATUS

approved

A209544

Primes not expressed in form n<+>2, where operation <+> defined in A206853.

+10
4

3, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Trivially every odd prime is expressed in form n<+>1 (cf. A208982).

Are these related to A141174, A045390 or A007519? - R. J. Mathar, Mar 13 2012

LINKS

Table of n, a(n) for n=1..51.

FORMULA

For n>=2, a(n) = A007519(n-1). - Vladimir Shevelev, Apr 18 2012

CROSSREFS

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A007519, A209554, A182187.

KEYWORD

nonn,base

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Mar 10 2012

STATUS

approved

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).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..55.

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.

EXAMPLE

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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..53.

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.

EXAMPLE

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).

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Colin Barker, Apr 04 2015

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

STATUS

approved

A141778

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

Is this the same as A038977? - R. J. Mathar, Jul 04 2008

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..55.

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.

EXAMPLE

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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

STATUS

approved

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