A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.
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
A107008 Primes of the form x^2 + 24*y^2.
73, 97, 193, 241, 313, 337, 409, 433, 457, 577, 601, 673, 769, 937, 1009, 1033, 1129, 1153, 1201, 1249, 1297, 1321, 1489, 1609, 1657, 1753, 1777, 1801, 1873, 1993, 2017, 2089, 2113, 2137, 2161, 2281, 2377, 2473, 2521, 2593, 2617, 2689, 2713
Offset: 1
Comments
Presumably this is the same as primes congruent to 1 mod 24, so a(n) = 24*A111174(n) + 1. - N. J. A. Sloane, Jul 11 2008. Checked for all terms up to 2 million. - Vladimir Joseph Stephan Orlovsky, May 18 2011.
Discriminant = -96.
Primes of the quadratic form are a subset of the primes congruent to 1 (mod 24). [Proof. For 0 <= x, y <= 23, the only values mod 24 that x^2 + 24*y^2 can take are 0, 1, 4, 9, 12 or 16. All of these r except 1 have gcd(r, 24) > 1 so if x^2 + 24*y^2 is prime its remainder mod 24 must be 1.] - David A. Corneth, Jun 08 2020
More advanced mathematics seems to be needed to determine whether this sequence lists all primes congruent to 1 (mod 24). Note the significance of 24 being a convenient number, as described in A000926. See also Sloane et al., Binary Quadratic Forms and OEIS, which explains how the table in A139642 may be used for this determination. - Peter Munn, Jun 21 2020
Primes == 1 (mod 2^3*3) are the intersection of the primes == 1 (mod 2^3) in A007519 and the primes == 1 (mod 3) in A002476, by the Chinese remainder theorem. - R. J. Mathar, Jun 11 2020
Links
- Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 143 terms from N. J. A. Sloane]
- P. L. Clark, J. Hicks, H. Parshall, K. Thompson, GONI: primes represented by binary quadratic forms, INTEGERS 13 (2013) #A37
- D. A. Cox, Primes of the form x^2 + n*y^2, A Wiley-Interscience publication, 1989
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
- J. Voight, Quadratic forms that represent almost the same primes, Math. Comp. 76 (2007) 1589-1617
Crossrefs
Programs
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Mathematica
QuadPrimes[1, 0, 24, 10000] (* see A106856 *)
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PARI
is(n) = isprime(n) && #qfbsolve(Qfb(1, 0, 24), n) == 2 \\ David A. Corneth, Jun 21 2020
Extensions
Recomputed b-file, deleted incorrect Mma program. - N. J. A. Sloane, Jun 08 2014
A155716 Numbers of the form N = a^2 + 6b^2 for some positive integers a,b.
7, 10, 15, 22, 25, 28, 31, 33, 40, 42, 49, 55, 58, 60, 63, 70, 73, 79, 87, 88, 90, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 145, 150, 151, 154, 159, 160, 166, 168, 175, 177, 186, 193, 196, 198, 199, 202, 214, 217, 220, 223, 225, 231, 232, 240
Offset: 1
Comments
Subsequence of A002481 (which allows for a and b to be zero).
Primes are in A033199. - Bernard Schott, Sep 20 2019
Links
- David A. Corneth, Table of n, a(n) for n = 1..17743 (terms <= 10^5)
Crossrefs
Programs
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Mathematica
With[{upto=240},Select[Union[#[[1]]^2+6#[[2]]^2&/@Tuples[ Range[Sqrt[ upto]], 2]],#<=upto&]] (* Harvey P. Dale, Aug 05 2016 *) -
PARI
isA155716(n,/* optional 2nd arg allows us to get other sequences */c=6) = { for(b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))} for( n=1,999, isA155716(n) & print1(n","))
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PARI
upto(n) = my(res=List()); for(i=1,sqrtint(n),for(j=1, sqrtint((n - i^2) \ 6), listput(res, i^2 + 6*j^2))); listsort(res,1); res \\ David A. Corneth, Sep 18 2019
A002481 Numbers of form x^2 + 6y^2.
0, 1, 4, 6, 7, 9, 10, 15, 16, 22, 24, 25, 28, 31, 33, 36, 40, 42, 49, 54, 55, 58, 60, 63, 64, 70, 73, 79, 81, 87, 88, 90, 96, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 144, 145, 150, 151, 154, 159, 160, 166, 168, 169, 175, 177, 186, 193, 196, 198, 199, 202, 214
Offset: 1
Keywords
Comments
Norms of numbers in Z[sqrt(-6)]. - Alonso del Arte, Sep 23 2014
It seems that a positive integer n is in this sequence if and only if the p-adic order ord_p(n) of n is even for any prime p with floor(p/12) odd, and the number of prime divisors p == 5 or 11 (mod 24) with ord_p(n) odd has the same parity with ord_2(n) + ord_3(n). - Zhi-Wei Sun, Mar 24 2018
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Zak Seidov, Table of n, a(n) for n = 1..2064 (terms <= 10000).
- Leonhard Euler, E388 Vollständige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 425.
Programs
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Maple
N:= 10^4: # to get all terms <= N {seq(seq(a^2 + 6*b^2, a = 0 .. floor(sqrt(N-6*b^2))), b = 0 .. floor(sqrt(N/6)))}; # for Maple 11, or earlier, uncomment the next line # sort(convert(%,list)); # Robert Israel, Sep 24 2014 -
Mathematica
lim = 10^4; k = 6; Union@Flatten@Table[x^2 + k * y^2, {y, 0, Sqrt[lim/k]}, {x, 0, Sqrt[lim - k * y^2]}] (* Zak Seidov, Mar 30 2011 *)
A107006 Primes of the form 4x^2-4xy+7y^2, with x and y nonnegative.
7, 31, 79, 103, 127, 151, 199, 223, 271, 367, 439, 463, 487, 607, 631, 727, 751, 823, 919, 967, 991, 1039, 1063, 1087, 1231, 1279, 1303, 1327, 1399, 1423, 1447, 1471, 1543, 1567, 1663, 1759, 1783, 1831, 1879, 1951, 1999, 2143, 2239, 2287, 2311
Offset: 1
Comments
Discriminant=-96.
Also, primes of the form 24n+7. - Artur Jasinski, Nov 25 2007 [See the Reble link]
Also primes of the forms 4x^2+4xy+7y^2, 7x^2+6xy+15y^2, 7x^2+2xy+7y^2 and 7x^2+4xy+28y^2. See A140633. - T. D. Noe, May 19 2008
Also, primes of form u^2+6v^2 with odd v while sequence A107008 is even v. This can be seen by expressing its form as (2x-y)^2+6y^2 (where y can only be odd) while the latter is x^2+6(2y)^2. Additionally, this sequence is 7 mod 24 while the second is 1 mod 24 and together, they are the primes of form x^2+6y^2 (A033199) which are either {1,7} mod 24. - Tito Piezas III, Jan 01 2009
Links
- Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 168 terms from N. J. A. Sloane]
- Don Reble, Notes on this sequence
- J. Liouville, Théorème concernant les nombres premiers de la forme 24µ + 7, Journal de mathématiques pures et appliquées 2e série, tome 4 (1859), pp. 399-400.
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Crossrefs
Cf. A124477.
Programs
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Mathematica
a = {}; Do[If[PrimeQ[24n + 7], AppendTo[a, 24n + 7]], {n, 0, 100}]; a (* Artur Jasinski, Nov 25 2007 *) QuadPrimes2[4, -4, 7, 10000] (* see A106856 *) Select[24*Range[0,4000]+7,PrimeQ] (* Harvey P. Dale, May 13 2018 *)
Extensions
Recomputed b-file and deleted erroneous Mma program by N. J. A. Sloane, Jun 08 2014
A155712 Intersection of A092572 and A155716: N = a^2 + 3b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.
7, 28, 31, 49, 63, 73, 79, 97, 100, 103, 112, 124, 127, 151, 175, 193, 196, 199, 217, 223, 241, 252, 271, 279, 292, 313, 316, 337, 343, 367, 388, 400, 409, 412, 433, 439, 441, 448, 457, 463, 484, 487, 496, 508, 511, 553, 567, 577, 601, 604, 607, 631, 657, 673
Offset: 1
Comments
From Robert Israel, Jan 19 2025: (Start)
If k is a term, then so is j^2 * k for all positive integers j.
The primes in this sequence appear to be A033199.
(End)
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Maple
N:= 1000: # for terms <= N A:= {seq(seq(a^2 + 3*b^2, b=1 .. floor(sqrt((N-a^2)/3))),a=1..floor(sqrt(N)))} intersect {seq(seq(c^2 + 6*d^2, d = 1 .. floor(sqrt((N-c^2)/6))),c=1..floor(sqrt(N)))}: sort(convert(A,list)); # Robert Israel, Jan 19 2025 -
PARI
isA155712(n,/* optional 2nd arg allows to get other sequences */c=[6,3]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) && next(2)); return);1} for( n=1,999, isA155712(n) && print1(n",")) \\ Update to modern PARI syntax (& -> &&) by M. F. Hasler, Jan 18 2025
A216509 Primes which cannot be written in the form a^2 + 6*b^2.
2, 3, 5, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 83, 89, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 191, 197, 211, 227, 229, 233, 239, 251, 257, 263, 269, 277, 281, 283, 293, 307, 311, 317, 331, 347, 349, 353
Offset: 1
Keywords
Comments
These are primes congruent to {5, 11, 13, 17, 19, 23} mod 24.
Crossrefs
Cf. A033199.
Comments
References
Links
Crossrefs
Programs
Mathematica
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 *)PARI
Extensions