A007521 -id:A007521 - cp's OEIS frontend

A230076 a(n) = (A007521(n)-1)/4.

1, 3, 7, 9, 13, 15, 25, 27, 37, 39, 43, 45, 49, 57, 67, 69, 73, 79, 87, 93, 97, 99, 105, 115, 127, 135, 139, 153, 163, 165, 169, 175, 177, 183, 189, 193, 199, 205, 207, 213, 219, 235, 249, 253, 255, 265, 267, 273, 277, 279, 295, 303, 307

Offset: 1

Wolfdieter Lang, Oct 24 2013

Because A007521(n) are the primes congruent 5 (mod 8) it is clear that a(n) is congruent 1 (mod 2), that is odd.

2*a(n) = A055034(A007521(n)), the degree of the minimal polynomial C(A007521(n), x) of 2*rho(Pi/A007521(n)) (see A187360).

			The minimal polynomial C(A007521(2), x) = C(13, x) has degree 6 = 2*a(2) because C(13, x) = x^6 - x^5 - 5*x^4 + 4*x^3 + 6*x^2 - 3*x -1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007521, A055034, A187360, 4*A005123 (1 (mod 8) case), A186287 (3 (mod 8) case), A186302 (7 (mod 8) case).

(Select[8*Range[0, 200] + 5, PrimeQ] - 1)/4 (* Amiram Eldar, Jun 08 2022 *)

a(n) = (A007521(n)-1)/4.

A095011 Number of 8k+5 primes (A007521) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 1, 1, 3, 2, 6, 11, 20, 35, 63, 118, 216, 407, 759, 1424, 2686, 5094, 9652, 18472, 35068, 67068, 128421, 246403, 473647, 911479, 1757076, 3390256, 6552075, 12675294, 24545280, 47583568, 92332958, 179313476, 348551899, 678031524, 1319932086, 2571422341

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. A007521, A091128, A095010, A095014.

Module[{nn = 21, s}, s = Select[8 Range[0, 2^nn] + 5, PrimeQ]; Table[Count[s, ?(2^n < # < 2^(n + 1) &)], {n, nn}]] (* _Michael De Vlieger, Jul 23 2017 *)

a(n) = A095014(n) - A095010(n).

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

A096634 Let p = n-th prime == 5 (mod 8) (A007521); a(n) = smallest prime q such that p is not a square mod q.

Original entry on oeis.org

3, 5, 3, 5, 3, 7, 3, 11, 3, 5, 3, 7, 3, 7, 3, 5, 3, 3, 7, 5, 3, 5, 13, 3, 3, 11, 3, 5, 3, 7, 3, 3, 13, 5, 5, 3, 3, 3, 7, 5, 5, 3, 5, 3, 7, 3, 7, 5, 3, 5, 3, 5, 3, 5, 3, 3, 3, 11, 11, 5, 3, 13, 5, 3, 17, 3, 7, 5, 3, 3, 7, 11, 7, 3, 3, 5, 3, 3, 3, 7, 5, 3, 3, 3, 11, 3, 13, 5, 3, 3, 7, 3, 3, 11, 5, 3, 3, 5, 3

Offset: 1

Robert G. Wilson v, Jun 24 2004

nonn

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

Cf. A094928, A096633, A096635, A096638.

g:= proc(n) local p;
  p:= 1;
  do
    p:= nextprime(p);
    if numtheory:-quadres(n,p) = -1 then return p fi
  od
end proc:
map(g, select(isprime, [seq(i,i=5..10000,8)])); # Robert Israel, Apr 17 2023

f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; f /@ Select[ Prime[ Range[435]], Mod[ #, 8] == 5 &]

A096639 Smallest prime p == 5 mod 8 (A007521) and p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

Original entry on oeis.org

5, 13, 61, 109, 421, 1621, 7309, 8941, 13381, 82021, 365509, 300301, 1336141, 644869, 8658589, 3462229, 6810301, 16145221, 165163909, 43030381, 163384621, 249623581, 2283397141, 1272463669, 2055693949

Offset: 0

Robert G. Wilson v, Jun 24 2004

nonn

Same as smallest prime p == 5 mod 8 with the property that the Legendre symbol (p|q) = 1 for the first n odd primes q = prime(k+1), k = 1, 2, ..., n, and (p|q) = -1 for q = prime(n+2).

Cf. A096634, A096636.

f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; t = Table[0, {50}]; Do[p = Prime[n]; If[Mod[p, 8] == 5, a = f[p]; If[ t[[ PrimePi[a]]] == 0, t[[ PrimePi[a]]] = p; Print[ PrimePi[a], " = ", p]]], {n, 10^9}]; t

Better name from Jonathan Sondow, Mar 07 2013

A096640 Smallest prime p == 7 mod 8 (A007521) and p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

Original entry on oeis.org

23, 7, 31, 79, 631, 751, 2311, 21319, 48799, 82471, 256279, 78439, 1768831, 1365079, 2631511, 1427911, 4355311, 5715319, 49196359, 117678031, 180628639, 475477759, 452980999

Offset: 0

Robert G. Wilson v, Jun 24 2004

nonn

Same as smallest prime p == 7 mod 8 with the property that the Legendre symbol (p|q) = 1 for the first n odd primes q = prime(k+1), k = 1, 2, ..., n, and (p|q) = -1 for q = prime(n+2).

Cf. A096635, A096636.

f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; t = Table[0, {50}]; Do[p = Prime[n]; If[Mod[p, 8] == 7, a = f[p]; If[ t[[ PrimePi[a]]] == 0, t[[ PrimePi[a]]] = p; Print[ PrimePi[a], " = ", p]]], {n, 10^9}]; t

Better name from Jonathan Sondow, Mar 07 2013

A157115 Alternate terms of A007519, A007520, A007521, A007522.

Original entry on oeis.org

17, 3, 5, 7, 41, 11, 13, 23, 73, 19, 29, 31, 89, 43, 37, 47, 97, 59, 53, 71, 113, 67, 61, 79, 137, 83, 101, 103, 193, 107, 109, 127, 233, 131, 149, 151, 241, 139, 157, 167, 257, 163, 173, 191, 281, 179, 181, 199, 313, 211, 197, 223, 337, 227, 229, 239, 353, 251, 269

Offset: 1

Zak Seidov and N. J. A. Sloane, Feb 23 2009

Or, read the following table by columns:
17,41,73,89,97,113,137,193,233,241,257,281,313,337,353,401,409,... (primes = = 1 mod 8)
3,11,19,43,59,67,83,107,131,139,163,179,211,227,251,283,307,331,... (primes == 3 mod 8)
5,13,29,37,53,61,101,109,149,157,173,181,197,229,269,277,293,317,... (primes == 5 mod 8)
7,23,31,47,71,79,103,127,151,167,191,199,223,239,263,271,311,359,... (primes == 7 mod 8)

			The first four primes congruent to (1,3,5,7) mod 8 are 17,3,5,7, hence a(1..4)=17,3,5,7;
The next four primes congruent to (1,3,5,7) mod 8 are 41,11,13,23, hence a(5..8)=41,11,13,23, etc.

Zak Seidov, Table of n, a(n) for n = 1..4000

Cf. A007519 - A007522, A042987.

s[i_]:=(c=0;a=2*i-1;Reap[Do[If[PrimeQ[a],c++;Sow[a]];If[c>99,Break[],a = a+8],{10^8}]][[2,1]]);Flatten[Transpose[Table[s[i],{i,4}]]]; (* Zak Seidov, Jan 16 2013 *)

A186299 (A007521(n)-1)/2.

Original entry on oeis.org

2, 6, 14, 18, 26, 30, 50, 54, 74, 78, 86, 90, 98, 114, 134, 138, 146, 158, 174, 186, 194, 198, 210, 230, 254, 270, 278, 306, 326, 330, 338, 350, 354, 366, 378, 386, 398, 410, 414, 426, 438, 470, 498, 506, 510, 530, 534, 546, 554, 558, 590, 606

Offset: 1

Marco Matosic, Feb 17 2011

nonn

a(n) = A186300(n)-1.

A186300 (A007521(n)+1)/2.

Original entry on oeis.org

3, 7, 15, 19, 27, 31, 51, 55, 75, 79, 87, 91, 99, 115, 135, 139, 147, 159, 175, 187, 195, 199, 211, 231, 255, 271, 279, 307, 327, 331, 339, 351, 355, 367, 379, 387, 399, 411, 415, 427, 439, 471, 499, 507, 511, 531, 535, 547, 555, 559, 591, 607

Offset: 1

Marco Matosic, Feb 17 2011

nonn

a(n) = A186299(n)+1.

A186301 a(n) = A007521(n) - 2.

Original entry on oeis.org

3, 11, 27, 35, 51, 59, 99, 107, 147, 155, 171, 179, 195, 227, 267, 275, 291, 315, 347, 371, 387, 395, 419, 459, 507, 539, 555, 611, 651, 659, 675, 699, 707, 731, 755, 771, 795, 819, 827, 851, 875, 939, 995, 1011, 1019, 1059, 1067, 1091, 1107, 1115, 1179, 1211

Offset: 1

Marco Matosic, Feb 17 2011

nonn

Cf. A007521.

a(52) corrected by Georg Fischer, Jun 27 2020

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

Original entry on oeis.org

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821

Offset: 1

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

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.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
QuadPrimes2[1, 1, 2, 1000]
(This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014)
max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

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cp's OEIS Frontend

A230076 a(n) = (A007521(n)-1)/4.

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A095011 Number of 8k+5 primes (A007521) in range ]2^n,2^(n+1)].

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A096634 Let p = n-th prime == 5 (mod 8) (A007521); a(n) = smallest prime q such that p is not a square mod q.

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A096639 Smallest prime p == 5 mod 8 (A007521) and p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

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A096640 Smallest prime p == 7 mod 8 (A007521) and p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

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A157115 Alternate terms of A007519, A007520, A007521, A007522.

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A186299 (A007521(n)-1)/2.

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A186300 (A007521(n)+1)/2.

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A186301 a(n) = A007521(n) - 2.

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A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

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