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
A341789 Norms of prime elements in Z[(1+sqrt(-67))/2], the ring of integers of Q(sqrt(-67)).
4, 9, 17, 19, 23, 25, 29, 37, 47, 49, 59, 67, 71, 73, 83, 89, 103, 107, 121, 127, 131, 149, 151, 157, 163, 167, 169, 173, 181, 193, 199, 211, 223, 227, 241, 257, 263, 269, 277, 283, 293, 307, 317, 349, 359, 389, 397, 419, 421, 431, 439, 449, 457, 461
Offset: 1
Comments
Also norms of prime ideals in Z[(1+sqrt(-67))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes such that (p,67) >= 0 and the squares of primes such that (p,67) = -1, where (p,67) is the Legendre symbol.
For primes p such that (p,67) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-67))/2], namely (x + y*(1+sqrt(-67))/2) and (x + y*(1-sqrt(-67))/2), where (x,y) is a solution to x^2 + x*y + 17*y^2 = p; for p = 67, (sqrt(-67)) is the unique ideal with norm p; for primes p with (p,67) = -1, (p) is the only ideal with norm p^2.
Examples
norm((1 + sqrt(-67))/2) = norm((1 - sqrt(-67))/2) = 17; norm((3 + sqrt(-67))/2) = norm((3 - sqrt(-67))/2) = 19; norm((5 + sqrt(-67))/2) = norm((5 - sqrt(-67))/2) = 23; norm((7 + sqrt(-67))/2) = norm((7 - sqrt(-67))/2) = 29; ... norm((31 + sqrt(-67))/2) = norm((31 - sqrt(-67))/2) = 257.
Links
- Jianing Song, Table of n, a(n) for n = 1..10000
Crossrefs
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A318982.
The total number of elements with norm n is given by A318984.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), this sequence (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.
Programs
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PARI
isA341783(n) = my(disc=-67); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
A373751 Array read by ascending antidiagonals: p is a term of row A(n) if and only if p is a prime and p is a quadratic residue modulo prime(n).
2, 3, 3, 5, 7, 5, 2, 11, 13, 7, 3, 7, 19, 19, 11, 3, 5, 11, 29, 31, 13, 2, 13, 11, 23, 31, 37, 17, 5, 13, 17, 23, 29, 41, 43, 19, 2, 7, 17, 23, 31, 37, 59, 61, 23, 5, 3, 11, 19, 29, 37, 43, 61, 67, 29, 2, 7, 13, 17, 43, 43, 47, 53, 71, 73, 31, 3, 5, 13, 23, 19, 47, 53, 53, 67, 79, 79, 37
Offset: 1
Comments
p is a term of A(n) <=> p is prime and there exists an integer q such that q^2 is congruent to p modulo prime(n).
Examples
Note that the cross-references are hints, not assertions about identity. . [ n] [ p] [ 1] [ 2] [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... A000040 [ 2] [ 3] [ 3, 7, 13, 19, 31, 37, 43, 61, 67, 73, ... A007645 [ 3] [ 5] [ 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, ... A038872 [ 4] [ 7] [ 2, 7, 11, 23, 29, 37, 43, 53, 67, 71, ... A045373 [ 5] [11] [ 3, 5, 11, 23, 31, 37, 47, 53, 59, 67, ... A056874 [ 6] [13] [ 3, 13, 17, 23, 29, 43, 53, 61, 79, 101, .. A038883 [ 7] [17] [ 2, 13, 17, 19, 43, 47, 53, 59, 67, 83, ... A038889 [ 8] [19] [ 5, 7, 11, 17, 19, 23, 43, 47, 61, 73, ... A106863 [ 9] [23] [ 2, 3, 13, 23, 29, 31, 41, 47, 59, 71, ... A296932 [10] [29] [ 5, 7, 13, 23, 29, 53, 59, 67, 71, 83, ... A038901 [11] [31] [ 2, 5, 7, 19, 31, 41, 47, 59, 67, 71, ... A267481 [12] [37] [ 3, 7, 11, 37, 41, 47, 53, 67, 71, 73, ... A038913 [13] [41] [ 2, 5, 23, 31, 37, 41, 43, 59, 61, 73, ... A038919 [14] [43] [11, 13, 17, 23, 31, 41, 43, 47, 53, 59, ... A106891 [15] [47] [ 2, 3, 7, 17, 37, 47, 53, 59, 61, 71, ... A267601 [16] [53] [ 7, 11, 13, 17, 29, 37, 43, 47, 53, 59, ... A038901 [17] [59] [ 3, 5, 7, 17, 19, 29, 41, 53, 59, 71, ... A374156 [18] [61] [ 3, 5, 13, 19, 41, 47, 61, 73, 83, 97, ... A038941 [19] [67] [17, 19, 23, 29, 37, 47, 59, 67, 71, 73, ... A106933 [20] [71] [ 2, 3, 5, 19, 29, 37, 43, 71, 73, 79, ... [21] [73] [ 2, 3, 19, 23, 37, 41, 61, 67, 71, 73, ... A038957 [22] [79] [ 2, 5, 11, 13, 19, 23, 31, 67, 73, 79, ... [23] [83] [ 3, 7, 11, 17, 23, 29, 31, 37, 41, 59, ... [24] [89] [ 2, 5, 11, 17, 47, 53, 67, 71, 73, 79, ... A038977 [25] [97] [ 2, 3, 11, 31, 43, 47, 53, 61, 73, 79, ... A038987 . Prime(n) is a term of row n because for all n >= 1, n is a quadratic residue mod n.
Links
- Robert G. Wilson v, Table of n, a(n) for n = 1..10011 (the first 141 antidiagonals, flattened).
Crossrefs
Programs
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Maple
A := proc(n, len) local c, L, a; a := 2; c := 0; L := NULL; while c < len do if NumberTheory:-QuadraticResidue(a, n) = 1 and isprime(a) then L := L,a; c := c + 1 fi; a := a + 1 od; [L] end: seq(print(A(ithprime(n), 10)), n = 1..25);
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Mathematica
f[m_, n_] := Block[{p = Prime@ m}, Union[ Join[{p}, Select[ Prime@ Range@ 22, JacobiSymbol[#, If[m > 1, p, 1]] == 1 &]]]][[n]]; Table[f[n, m -n +1], {m, 12}, {n, m, 1, -1}] (* To read the array by descending antidiagonals, just exchange the first argument with the second in the function "f" called by the "Table"; i.e., Table[ f[m -n +1, n], {m, 12}, {n, m, 1, -1}] *) -
PARI
A373751_row(n, LIM=99)={ my(q=prime(n)); [p | p <- primes([1,LIM]), issquare( Mod(p, q))] } \\ M. F. Hasler, Jun 29 2024
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SageMath
# The function 'is_quadratic_residue' is defined in A373748. def A373751_row(n, len): return [a for a in range(len) if is_quadratic_residue(a, n) and is_prime(a)] for p in prime_range(99): print([p], A373751_row(p, 100))
A152053 a(n) = A144433(3n+1) + A144433(3n+2) + A144433(3n+3).
27, 36, 81, 72, 135, 108, 189, 144, 243, 180, 297, 216, 351, 252, 405, 288, 459, 324, 513, 360, 567, 396, 621, 432, 675, 468, 729, 504, 783, 540, 837, 576, 891, 612, 945, 648, 999, 684, 1053, 720, 1107
Offset: 0
Comments
All terms are multiples of 9.
Links
- Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, -1).
Programs
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Mathematica
Table[(9/2)(5 + (-1)^n)(n + 1), {n, 0, 40}] (* Jean-François Alcover, Feb 02 2019 *) LinearRecurrence[{0,2,0,-1},{27,36,81,72},50] (* Harvey P. Dale, Nov 07 2019 *) -
PARI
a(n) = 45*(n+1)/2 + 9*(-1)^n*(n+1)/2 \\ Jianing Song, Feb 04 2019
Formula
From R. J. Mathar, May 21 2009: (Start)
G.f.: 9*(3+4*x+3*x^2)/((x-1)^2*(1+x)^2).
a(n) = 45*(n+1)/2 + 9*(-1)^n*(n+1)/2. (End)
a(n+4) = 2*a(n+2) - a(n). - Jianing Song, Feb 04 2019
Extensions
Edited by R. J. Mathar, May 21 2009
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