A014754 -id:A014754 - cp's OEIS frontend

A107132 Primes of the form 2x^2 + 13y^2.

2, 13, 31, 149, 167, 317, 359, 397, 463, 487, 509, 613, 661, 709, 839, 1061, 1087, 1103, 1151, 1181, 1367, 1471, 1783, 1789, 1861, 2039, 2111, 2221, 2269, 2437, 2503, 2621, 2647, 2917, 2927, 2957, 3023, 3079, 3167, 3229, 3373, 3541, 3853

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -104. Binary quadratic forms ax^2+cy^2 have discriminant d=-4ac. We consider sequences of primes produced by forms with -400<=d<=0, a<=c and gcd(a,c)=1. These restrictions yield 173 sequences of prime numbers, which are organized by discriminant below. See A106856 for primes of the form ax^2+bxy+cy^2 with discriminant > -100.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A033218 (d=-104), A014752 (d=-108), A107133, A107134 (d=-112), A033219 (d=-116), A107135-A107137, A033220 (d=-120), A033221 (d=-124), A105389 (d=-128), A107138, A033222 (d=-132), A107139, A033223 (d=-136), A107140, A033224 (d=-140), A107141, A107142 (d=-144), A033225 (d=-148), A107143, A033226 (d=-152), A033227 (d=-156), A107144, A107145 (d=-160), A033228 (d=-164), A107146-A107148, A033229 (d=-168).
Cf. A033230 (d=-172), A107149, A107150 (d=-176), A107151, A107152 (d=-180), A107153, A033231 (d=-184), A033232 (d=-188), A141373 (d=-192), A107155 (d=-196), A107156, A107157 (d=-200), A107158, A033233 (d=-204), A107159, A107160 (d=-208), A033234 (d=-212), A107161, A107162 (d=-216), A033235 (d=-220), A107163, A107164 (d=-224), A107165, A033236 (d=-228), A107166, A033237 (d=-232), A033238 (d=-236).
Cf. A107167-A107169 (d=-240), A033239 (d=-244), A107170, A033240 (d=-248), A014754 (d=-256), A107171, A033241 (d=-260), A107172-A107174, A033242 (d=-264), A033243 (d=-268), A107175, A107176 (d=-272), A107177, A033244 (d=-276), A107178-A107180, A033245 (d=-280), A033246 (d=-284), A107181 (d=-288), A033247 (d=-292), A107182, A033248 (d=-296), A107183, A107184 (d=-300), A107185, A107186 (d=-304), A107187, A033249 (d=-308).
Cf. A107188-A107190, A033250 (d=-312), A033251 (d=-316), A107191, A107192 (d=-320), A107193 (d=-324), A107194, A033252 (d=-328), A033253 (d=-332), A107195-A107198 (d=-336), A107199, A033254 (d=-340), A107200, A033255 (d=-344), A033256 (d=-348), A107132 A107201, A107202 (d=-352), A033257 (d=-356), A107203-A107206 (d=-360), A107207, A033258 (d=-364), A107208, A107209 (d=-368), A107210, A033202 (d=-372).
Cf. A107211, A033204 (d=-376), A033206 (d=-380), A107212, A107213 (d=-384), A033208 (d=-388), A107214, A107215 (d=-392), A107216, A107217 (d=-396), A107218, A107219 (d=-400).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[2, 0, 13, 10000] (* see A106856 *)

list(lim)=my(v=List([2,13]),t); for(y=1,sqrtint(lim\13), for(x=1,sqrtint((lim-13*y^2)\2), if(isprime(t=2*x^2+13*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 07 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

A070179 Primes p such that x^2 = 2 has a solution mod p, but x^(2^2) = 2 has no solution mod p.

Original entry on oeis.org

17, 41, 97, 137, 193, 241, 313, 401, 409, 433, 449, 457, 521, 569, 641, 673, 761, 769, 809, 857, 929, 953, 977, 1009, 1129, 1297, 1321, 1361, 1409, 1489, 1657, 1697, 1873, 1993, 2017, 2081, 2137, 2153, 2161, 2297, 2377, 2417, 2521, 2609, 2617, 2633, 2713

Offset: 1

Klaus Brockhaus, Apr 29 2002

Complement of A014754 with regard to primes of the form 8*k+1.
These appear to be the primes p for which 4^((p-1)*n/8) mod p = (p-2)*( n mod 2)+1. For example, 4^(5*n) mod 41 = 1,40,1,40,1,40...= 39*(n mod 2)+1 and 4^(30*n) mod 241 = 1,240,1,240,1,240...= 239*(n mod 2) +1. - Gary Detlefs, Jul 06 2014
Primes p == 1 mod 8 such that 2^((p-1)/4) == -1 mod p. - Robert Israel, Jul 06 2014
A very similar sequence is A293394. - Jonas Kaiser, Nov 08 2017

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A038873, A040098, A040100, A059667, A070180-A070188, A014754.

[p: p in PrimesUpTo(3000) | not exists{x: x in ResidueClassRing(p) | x^4 eq 2} and exists{x: x in ResidueClassRing(p) | x^2 eq 2}]; // Vincenzo Librandi, Sep 21 2012

select(p -> isprime(p) and 2 &^((p-1)/4) mod p = p-1, [8*k+1$k=1..10000]); # Robert Israel, Jul 06 2014

PARI
```
forprime(p=2,2720,x=0; while(x
				
```

{a(n) = local(m, c, x); if( n<1, 0, c = 0; m = 1; while( cMichael Somos, Mar 22 2008 */

ok(p, r, k1, k2)={
if ( Mod(r,p)^((p-1)/gcd(k1,p-1))!=1, return(0) );
if ( Mod(r,p)^((p-1)/gcd(k2,p-1))==1, return(0) );
return(1);
}
forprime(p=2,10^4, if (ok(p,2,2,2^2),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */

is(n)=n%8==1 && Mod(2,n)^(n\4)==-1 && isprime(n) \\ Charles R Greathouse IV, Nov 10 2017

Primes of the form 8*k + 1 but not x^2 + 64*y^2. - Michael Somos, Mar 22 2008

a(n) ~ 8n log n. - Charles R Greathouse IV, Nov 10 2017

A065909 First solution mod p of x^4 = 2 for primes p such that more than two solutions exist.

Original entry on oeis.org

18, 5, 27, 28, 35, 46, 131, 48, 252, 104, 45, 123, 51, 9, 69, 77, 51, 177, 472, 261, 55, 117, 224, 562, 12, 264, 273, 132, 127, 500, 17, 197, 107, 36, 206, 671, 127, 159, 137, 684, 329, 564, 316, 314, 197, 98, 661, 925, 461, 170, 930, 151, 1081, 333, 434, 924

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). There are integers which do occur thrice, e.g. 6624. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 27, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 27 is the least one.

Cf. A040098, A007522, A014754, A065910, A065911, A065912.

a065909(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>2,print1(s[1],",")))
a065909(4000)

a(n) = first (least) solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

Definition corrected by Harvey P. Dale, Apr 16 2015

A065910 Second solution mod p of x^4 = 2 for primes p such that more than two solution exists.

Original entry on oeis.org

25, 8, 47, 71, 46, 91, 158, 102, 278, 294, 216, 201, 355, 110, 297, 283, 161, 567, 490, 422, 578, 250, 309, 625, 344, 578, 287, 151, 164, 641, 736, 238, 474, 763, 408, 758, 406, 650, 813, 1090, 1043, 771, 328, 699, 902, 165, 857, 1000, 553, 1148, 1434, 955

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). In this section, there are no integers which do occur thrice. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 47, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 47 is the second one.

Cf. A040098, A007522, A014754, A065909, A065911, A065912.

a065910(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>2,print1(s[2],",")))
a065910(3500)

a(n) = second solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

A065911 Third solution mod p of x^4 = 2 for primes p such that more than two solution exists.

Original entry on oeis.org

48, 81, 66, 162, 211, 190, 179, 251, 299, 299, 385, 416, 526, 827, 736, 766, 936, 586, 703, 779, 639, 999, 980, 808, 1137, 975, 1314, 1458, 1557, 1112, 1041, 1563, 1415, 1150, 1681, 1355, 1723, 1623, 1468, 1303, 1398, 1702, 2265, 1958, 1787, 2668, 2000

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). In this section, there are no integers which do occur thrice. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 66, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 66 is the third one.

Cf. A040098, A007522, A014754, A065909, A065910, A065912.

a065911(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>2,print1(s[3],",")))
a065911(3000)

a(n) = third solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

A065912 Fourth solution mod p of x^4 = 2 for primes p such that more than two solution exists.

Original entry on oeis.org

55, 84, 86, 205, 222, 235, 206, 305, 325, 489, 556, 494, 830, 928, 964, 972, 1046, 976, 721, 940, 1162, 1132, 1065, 871, 1469, 1289, 1328, 1477, 1594, 1253, 1760, 1604, 1782, 1877, 1883, 1442, 2002, 2114, 2144, 1709, 2112, 1909, 2277, 2343, 2492, 2735

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three time in this sequence. Confirmed for the first 1182 terms of A014754 (primes < 100000). In this section, there are no integers which do occur thrice. Moreover, no integer is first, second, third or fourth solution for more than three primes. Confirmed for the first 2399 terms of A007522 and the first 1182 terms of A014754 (primes < 100000).

			a(3) = 86, since 113 is the third term of A014754, 27, 47, 66 and 86 are the solutions mod 113 of x^4 = 2 and 86 is the fourth one.

Cf. A040098, A007522, A014754, A065909, A065910, A065911.

a065912(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2]>3,print1(s[4],",")))
a065912(3000)

a(n) = fourth solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has more than two solutions mod p, i.e. p is the n-th term of A014754.

A351865 Primes of the form x^2 + 64*y^2 that divide some Fermat number.

Original entry on oeis.org

257, 65537, 2424833, 26017793, 63766529, 825753601, 1214251009, 6487031809, 2710954639361, 2748779069441, 6597069766657, 25991531462657, 76861124116481, 151413703311361, 1095981164658689, 1238926361552897, 1529992420282859521, 2663848877152141313, 3603109844542291969

Offset: 1

Arkadiusz Wesolowski, Apr 10 2022

nonn

A prime p = k*2^j + 1 (with k odd) belongs to this sequence if and only if p is a factor of a Fermat number 2^(2^m) + 1 for some m <= j - 3.

			a(1) = 1^2 + 64*2^2 = 257 is a prime factor of 2^(2^3) + 1;
a(2) = 1^2 + 64*32^2 = 65537 is a prime factor of 2^(2^4) + 1;
a(3) = 127^2 + 64*194^2 = 2424833 is a prime factor of 2^(2^9) + 1;
a(4) = 2047^2 + 64*584^2 = 26017793 is a prime factor of 2^(2^12) + 1;
a(5) = 7295^2 + 64*406^2 = 63766529 is a prime factor of 2^(2^12) + 1;

Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.

Cf. A014754, A023394, A228846.

isok(p) = if(p%8==1 && isprime(p), my(d=Mod(2, p)); d^((p-1)/4)==1 && d^2^valuation(p-1, 2)==1, return(0));

A014754 INTERSECT A023394.

cp's OEIS Frontend

A107132 Primes of the form 2x^2 + 13y^2.

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

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A070179 Primes p such that x^2 = 2 has a solution mod p, but x^(2^2) = 2 has no solution mod p.

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A065909 First solution mod p of x^4 = 2 for primes p such that more than two solutions exist.

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A065910 Second solution mod p of x^4 = 2 for primes p such that more than two solution exists.

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A065911 Third solution mod p of x^4 = 2 for primes p such that more than two solution exists.

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A065912 Fourth solution mod p of x^4 = 2 for primes p such that more than two solution exists.

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