Robin Visser - cp's OEIS frontend

A387207 The maximal norm of an additively indecomposable element in the real quadratic field Q(sqrt(D)), where D = A005117(n) is the n-th squarefree number.

2, 1, 1, 3, 2, 10, 5, 3, 2, 1, 4, 9, 1, 11, 2, 26, 7, 6, 10, 4, 2, 1, 9, 19, 13, 10, 7, 21, 9, 2, 25, 13, 9, 7, 58, 29, 15, 2, 16, 33, 33, 3, 14, 10, 18, 74, 1, 3, 2, 82, 41, 21, 43, 13, 22, 30, 7, 18, 5, 24, 25, 51, 34, 4, 106, 53, 27, 11, 37, 28, 57, 9, 59, 2, 122, 61, 42, 16, 130, 65, 11

Offset: 2

Robin Visser, Aug 21 2025

For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.

Let K = Q(sqrt(D)) be a real quadratic field. By studying the continued fraction expansion of sqrt(D), Dress and Scharlau classified all additively indecomposable elements of K and showed that every such indecomposable element has its norm bounded by the discriminant of K.

			For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) has norm either 1 or 2, thus a(2) = 2.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) has norm 1, thus a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) has norm 1, thus a(4) = 1.
For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) has norm either 1 or 3, thus a(5) = 3.

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.
Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.
Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Cf. A005117, A035015, A387203.

def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K. = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0, 2*s+1, 2) for t in range(cf[i+1])]
    return max([c.norm() for c in ind])

a(n) <= A005117(n) for all n >= 2 [Dress-Scharlau].

A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.

Original entry on oeis.org

2, 1, 1, 2, 2, 6, 3, 3, 2, 1, 5, 7, 1, 6, 2, 10, 5, 2, 8, 4, 2, 1, 7, 6, 4, 11, 2, 13, 8, 2, 7, 7, 4, 7, 20, 9, 11, 2, 9, 8, 19, 2, 6, 6, 21, 20, 1, 2, 2, 18, 9, 9, 16, 3, 21, 12, 3, 12, 2, 27, 11, 10, 18, 3, 34, 13, 17, 2, 8, 23, 12, 5, 18, 2, 22, 11, 24, 15, 26, 15, 6, 22, 27, 2, 31, 4, 2

Offset: 2

Robin Visser, Aug 21 2025

For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.

Let K = Q(sqrt(D)) be a real quadratic field. By studying the continued fraction expansion of sqrt(D), Dress and Scharlau classified all additively indecomposable elements of K and showed that every such indecomposable element has its norm bounded by the discriminant of K.

			For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) is of the form u or u*(2 + sqrt(2)), for some totally positive unit u. Thus a(2) = 2.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) is a totally positive unit, so a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) is a totally positive unit, so a(4) = 1.
For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) is of the form u or u*(3 + sqrt(6)), for some totally positive unit u. Thus a(5) = 2.

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.
Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.
Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Cf. A005117, A035015, A387207.

def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K. = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0,2*s+1,2) for t in range(cf[i+1])]
    for i in range(len(ind)):
        for j in range(i):
            if ((ind[i]/ind[j]) in OK) and (OK(ind[i]/ind[j]).is_unit()): break
        else: ans += 1
    return ans

A383643 Number of n-dimensional additively indecomposable positive definite integral lattices (or quadratic forms).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 2

Offset: 1

Robin Visser, May 09 2025

A positive definite integral lattice (or quadratic form) is additively indecomposable if it cannot be written as a sum of two nonzero positive semidefinite integral lattices (or quadratic forms). Any additively indecomposable lattice is also (orthogonally) indecomposable, although the converse need not hold.

By computing all additively indecomposable lattices of determinant up to 100, Opgenorth gave the lower bounds a(10) >= 7 and a(11) >= 13. Eisenbarth showed the lower bounds a(12) >= 29, a(13) >= 9, a(14) >= 10, a(15) >= 9, and a(16) >= 5.

			For n <= 8, the only n-dimensional additively indecomposable positive definite lattices are Z (of dimension 1), E6 (of dimension 6), E7 (of dimension 7), and E8 (of dimension 8).
For n = 9, the a(9) = 2 additively indecomposable rank 9 positive definite lattices were computed by Opgenorth. These are the two lattices with Gram matrices:
  [ 2  1  1  1  1  1  1  1  4]   [ 2  1  1  1  2  2  2  2  6]
  [ 1  2  1  1  1  1  1  1  4]   [ 1  2  1  1  2  2  2  2  6]
  [ 1  1  2  1  1  1  1  1  4]   [ 1  1  2  1  2  2  2  2  6]
  [ 1  1  1  2  1  1  1  1  4]   [ 1  1  1  2  2  2  2  2  6]
  [ 1  1  1  1  2  1  1  1  4]   [ 2  2  2  2  5  4  4  4 12]
  [ 1  1  1  1  1  2  1  1  4]   [ 2  2  2  2  4  5  4  4 12]
  [ 1  1  1  1  1  1  2  1  4]   [ 2  2  2  2  4  4  5  4 12]
  [ 1  1  1  1  1  1  1  2  4]   [ 2  2  2  2  4  4  4  5 12]
  [ 4  4  4  4  4  4  4  4 15],  [ 6  6  6  6 12 12 12 12 35],
  having determinant 7 and determinant 15 respectively.

Jürgen Opgenorth, Additiv unzerlegbare ganzzahlige quadratische Formen in den Dimensionen 9, 10 und 11. Diplomarbeit, RWTH Aachen, 1992.

Simon Eisenbarth, Additive Zerlegungen von Gittern, Bachelorarbeit, RWTH Aachen, 2014.
Paul Erdős and Chao Ko, On definite quadratic forms, which are not the sum of two definite or semi-definite forms, Acta Arith. 3, 102-122 (1938).
Louis J. Mordell, The representation of a definite quadratic form as a sum of two others, Ann. of Math. (2) 38 (1937), no. 4, 751-757.
Wilhelm Plesken, Additively indecomposable positive integral quadratic forms, J. Number Theory, 47 (1994), no. 3, 273-283.
Ruiqing Wang, Additively indecomposable positive definite integral lattices, Acta Math. Sin. (Engl. Ser.) 41 (2025), no. 3, 908-924.

Cf. A380746.

a(n) >= A380746(n).

A383067 The set of positive integers k which can be expressed as a sum of two units in some cyclic cubic field.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 19, 22

Offset: 1

Robin Visser, Apr 15 2025

These are all the positive integers k such that there exists some cubic number field K whose Galois group is cyclic (C_3) and contains units u, v in K such that k = u + v.

			For each positive integer k given in the sequence, it can be written as a sum of two units in some cyclic cubic field as follows:
1 = u + (-u+1), where u is a root of x^3 + x^2 - 2x - 1.
2 = u + (-u+2), where u is a root of x^3 - 3x - 1.
3 = (u^2) + (-u^2+3), where u is a root of x^3 + x^2 - 2x - 1.
4 = (u^2+2u) + (-u^2-2u+4), where u is a root of x^3 + x^2 - 2x - 1.
5 = (u^2-u) + (-u^2+u+5), where u is a root of x^3 + x^2 - 2x - 1.
7 = (u^2) + (-u^2+7), where u is a root of x^3 - x^2 - 4x - 1.
19 = (5u^2+9u) + (-5u^2-9u+19), where u is a root of x^3 + x^2 - 2x - 1.
22 = (4u^2-5u) + (-4u^2+5u+22), where u is a root of x^3 + x^2 - 2x - 1.

M. Tinková, R. Visser, and P. Yatsyna, Sums of two units in number fields, arXiv:2502.01345 [math.NT], 2025.
I. Vukusic and V. Ziegler, On a family of unit equations over simplest cubic fields, J. Théor. Nombres Bordeaux 34 (2022), no. 3, 705-718.

Cf. A383068.

A383068 The set of all integers k >= -1 with the property that there exist integers X and Y such that XY(X+Y) is nonzero and X^3 - kX^2Y - (k+3)XY^2 - Y^3 is a divisor of k^2 + 3*k + 9.

Original entry on oeis.org

-1, 0, 1, 2, 3, 5, 12, 54, 66, 1259, 2389

Offset: 1

Robin Visser, Apr 15 2025

Let F_k(X,Y) denote the polynomial X^3 - k*X^2*Y - (k+3)*X*Y^2 - Y^3. Then if there exist integers X,Y such that F_k(X,Y) is a divisor of k^2 + 3*k + 9, then F_{-k-3}(-Y, -X) is also a divisor of (-k-3)^2 + 3*(-k-3) + 9. Thus, to classify all integers k such that F_k(X,Y) divides k^2 + 3*k + 9 for some integers X and Y, it suffices to classify only those integers k where k >= -1.

Let L_k denote Shanks' simplest cubic field with defining polynomial x^3 - k*x^2 - (k+3)*x - 1. Then this sequence is also the set of all integers k >= -1 such that L_k is isomorphic to L_i for some i >= -1 not equal to k (as shown by Hoshi and Okazaki). In particular, we have the following isomorphisms between the simplest cubic fields: L_{-1} = L_5 = L_12 = L_1259, L_0 = L_3 = L_54, L_1 = L_66, and L_2 = L_2389.

			For each integer k >= -1 given in the sequence, the list of all pairs (X,Y) such that X*Y*(X+Y) is nonzero and such that F_k(X,Y) is a positive integer divisor of k^2 + 3k + 9 is as follows:
If k = -1, then (X,Y) = (-1,-1), (-1,2), (2,-1), (5,4), (4,-9), (-9,5), (2,1), (1,-3), or (-3,2).
If k = 0, then (X,Y) = (2,1), (1,-3), (-3, 2), (-1,-1), (-1, 2), or (2,-1).
If k = 1, then (X,Y) = (-5,-2), (-2, 7), or (7,-5).
If k = 2, then (X,Y) = (-7,-2), (-2, 9), or (9,-7).
If k = 3, then (X,Y) = (-1,-1), (-1, 2), (2,-1), (-4,-1), (-1, 5), or (5,-4).
If k = 5, then (X,Y) = (-1,-2), (-2, 3), (3,-1), (-4,-1), (-1, 5), (5,-4), (19, 3), (3,-22), or (-22, 19).
If k = 12, then (X,Y) = (-1,-1), (-1, 2), (2,-1), (-13,-1), (-1, 14), (14,-13), (-4,-1), (-1, 5), or (5,-4).
If k = 54, then (X,Y) = (-1,-2), (-2, 3), (3,-1), (-4,-1), (-1, 5), or (5,-4).
If k = 66, then (X,Y) = (-5,-2), (-2, 7), or (7,-5).
If k = 1259, then (X,Y) = (-4,-5), (-5, 9), (9,-4), (-13,-1), (-1, 14), (14,-13), (-3,-19), (-19, 22), or (22,-3).
If k = 2389, then (X,Y) = (-7,-2), (-2, 9), or (9,-7).

A. Hoshi, On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields, J. Number Theory 131 (2011), no. 11, 2135-2150. See Table 1 on page 2138.
T. Kashio and R. Sekigawa, The characterization of cyclic cubic fields with power integral bases, Kodai Math. J. 44 (2021), no. 2, 290-306.
R. Okazaki, The simplest cubic fields are non-isomorphic to each other, Doshisha University, Kyoto, Japan. [Cached copy at the Wayback Machine]
M. Tinková, R. Visser, and P. Yatsyna, Sums of two units in number fields, arXiv:2502.01345 [math.NT], 2025. See page 9.

Cf. A379019, A383067.

is_A383068 := function(k)
    R := PolynomialRing(Integers());
    T := Thue(x^3 - k*x^2 - (k+3)*x - 1);
    for d in Divisors(k^2 + 3*k + 9) do
        S := Solutions(T, d);
        for s in S do
            if (s[1]*s[2]*(s[1]+s[2]) ne 0) then return true; end if;
        end for;
    end for;
    return false;
end function;
[k : k in [-1..3000] | is_A383068(k)];

A380746 Number of n-dimensional indecomposable unimodular lattices (or quadratic forms).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 3, 11, 12, 27, 48, 176, 367, 1896, 14489, 356988

Offset: 1

Robin Visser, Jan 31 2025

The sequence {a(n)} is the inverse Euler transform of A005134.

King gives the lower bound a(29) >= 37563933 (using computations of Allombert--Chenevier).

			For n = 1, the only 1-dimensional indecomposable unimodular lattice is Z, thus a(1) = 1.
For n = 8, the only 8-dimensional indecomposable unimodular lattice is E8, thus a(8) = 1.
For n = 12, the only 12-dimensional indecomposable unimodular lattice is D12+, thus a(12) = 1.

Fu Zu Zhu, Construction of nondecomposable positive definite unimodular quadratic forms. Sci. Sinica Ser. A, 30 (1987), no. 1, 19-31.
Fu Zu Zhu, On nondecomposability and indecomposability of quadratic forms, Sci. Sinica Ser. A, 31 (1988), no. 3, 265-273.

Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
Etsuko Bannai, Positive definite unimodular lattices with trivial automorphism groups, Mem. Amer. Math. Soc., 85 (1990), no. 429, iv+70 pp.
Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third edition, Springer-Verlag, New York, 1999. lxxiv+703 pp.
Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863.
O. T. O'Meara, The construction of indecomposable positive definite quadratic forms, J. Reine Angew. Math., 276 (1975), 99-123.
Wilhelm Plesken, Additively indecomposable positive integral quadratic forms, J. Number Theory, 47 (1994), no. 3, 273-283.

Cf. A005134, A054907, A054909, A054911, A383643.

Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A005134(k)*x^k.

a(n) <= A054907(n) for all n > 1.

A380269 The minimal rank of an n-universal Z-lattice.

Original entry on oeis.org

4, 5, 6, 7, 8, 13, 15, 16, 28, 30

Offset: 1

Robin Visser, Jan 18 2025

a(n) is the least positive integer k such that there exists a positive definite Z-lattice of rank k which represents all positive definite Z-lattices of rank n.

Byeong-Kweon Oh gives the lower bound a(24) >= 6673.

			If n <= 5, then the diagonal lattice I_{n+3} is an n-universal Z-lattice of minimal rank, thus a(n) = n+3 for all n <= 5.

Daejun Kim and Byeong-Kweon Oh, Representations of finite number of quadratic forms with same rank, Ramanujan J., 56 (2021), no. 2, 631-644.
Byeong-Kweon Oh, Universal Z-lattices of minimal rank, Proc. Amer. Math. Soc., 128 (2000), no. 3, 683-689.
O. T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag Berlin Heidelberg, 1973.

Cf. A054911.

A379019 Positive integers k such that the simplest cubic field defined by x^3 - kx^2 - (k+3)x - 1 is not monogenic.

Original entry on oeis.org

21, 30, 41, 48, 57, 75, 84, 90, 100, 102, 103, 111, 129, 138, 139, 152, 154, 156, 165, 183, 188, 192, 201, 204, 210, 219, 235, 237, 246, 250, 264, 269, 271, 273, 291, 299, 300, 318, 327, 335, 345, 348, 354, 356, 372, 374, 381, 384, 398, 399, 404, 408, 426, 433, 435, 438, 446, 453, 462, 480

Offset: 1

Robin Visser, Dec 13 2024

nonn

These are the positive integers k such that the ring of integers O_K of the simplest cubic field K = Q[x]/(x^3 - k*x^2 - (k+3)*x - 1) does not have a power integral basis of the form {1, a, a^2} for any element a in O_K.

D. Gil-Muñoz and M. Tinková, Additive structure of non-monogenic simplest cubic fields, arXiv:2212.00364 [math.NT], 2022.
T. Kashio and R. Sekigawa, The characterization of cyclic cubic fields with power integral bases, Kodai Math. J. 44 (2021), no. 2, 290-306.
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.

Cf. A005472.

is_A379019 := function(k)
    R := PolynomialRing(Integers());
    K := NumberField(x^3 - k*x^2 - (k+3)*x - 1);
    return #IndexFormEquation(MaximalOrder(K), 1) eq 0;
end function;
[k : k in [1..1000] | is_A379019(k)];

A377607 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = 3 is solvable over the integers.

Original entry on oeis.org

1, 6, 13, 22, 33, 46, 61, 69, 73, 78, 94, 97, 109, 118, 141, 157, 166, 177, 181, 193, 213, 214, 222, 241, 249, 253, 262, 277, 286, 313, 321, 334, 337, 358, 366, 382, 393, 397, 409, 421, 429, 433, 438, 454, 457, 478, 481, 501, 502, 517, 526, 537, 541, 573, 598, 601, 613, 622, 649, 654, 661

Offset: 1

Robin Visser, Nov 02 2024

nonn

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

			The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[1, [2, 1]], [6, [3, 1]], [13, [4, 1]], [22, [5, 1]], [33, [6, 1]], [46, [7, 1]], [61, [8, 1]], [69, [108, 13]], [73, [94, 11]], [78, [9, 1]], [94, [223, 23]], [97, [10, 1]], [109, [9532, 913]], [118, [11, 1]], [141, [12, 1]], [157, [289580, 23111]], [166, [13, 1]], [177, [306, 23]], [181, [148, 11]], [193, [14, 1]], ...

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.
Eric Weisstein's World of Mathematics, Pell Equation.

Cf. A031396, A243655, A261246, A377600.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377607_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d,3)), count(max(startvalue,1)))
A377607_list = list(islice(A377607_gen(),61)) # Chai Wah Wu, Nov 03 2024

A377600 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -3 is solvable over the integers.

Original entry on oeis.org

1, 3, 4, 7, 12, 13, 19, 21, 28, 31, 39, 43, 52, 57, 61, 67, 73, 76, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 151, 157, 163, 172, 181, 183, 193, 199, 201, 211, 217, 228, 237, 241, 244, 247, 259, 268, 271, 273, 277, 283, 292, 301, 307, 309, 313, 327, 331, 337, 343, 364

Offset: 1

Robin Visser, Nov 02 2024

nonn

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

			The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[1, [1, 2]], [3, [0, 1]], [4, [1, 1]], [7, [2, 1]], [12, [3, 1]], [13, [7, 2]], [19, [4, 1]], [21, [9, 2]], [28, [5, 1]], [31, [11, 2]], [39, [6, 1]], [43, [13, 2]], [52, [7, 1]], [57, [15, 2]], [61, [5639, 722]], [67, [8, 1]], [73, [17, 2]], [76, [61, 7]], [84, [9, 1]], [91, [19, 2]], [93, [135, 14]], [97, [847, 86]], [103, [10, 1]], [109, [1399, 134]], [111, [21, 2]], [124, [11, 1]], [127, [293, 26]], [129, [159, 14]], [133, [23, 2]], [139, [224, 19]], [147, [12, 1]], [151, [86, 7]], [157, [25, 2]], [163, [932, 73]], [172, [13, 1]], [181, [11262809, 837158]], [183, [27, 2]], [193, [189743, 13658]], [199, [14, 1]], ...

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.
Eric Weisstein's World of Mathematics, Pell Equation.

Cf. A031396, A244819, A261246, A377607.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377600_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d,-3)), count(max(startvalue,1)))
A377600_list = list(islice(A377600_gen(),63)) # Chai Wah Wu, Nov 03 2024

cp's OEIS Frontend

User: Robin Visser

A387207 The maximal norm of an additively indecomposable element in the real quadratic field Q(sqrt(D)), where D = A005117(n) is the n-th squarefree number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

Formula

A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

A383643 Number of n-dimensional additively indecomposable positive definite integral lattices (or quadratic forms).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A383067 The set of positive integers k which can be expressed as a sum of two units in some cyclic cubic field.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A383068 The set of all integers k >= -1 with the property that there exist integers X and Y such that X*Y*(X+Y) is nonzero and X^3 - k*X^2*Y - (k+3)*X*Y^2 - Y^3 is a divisor of k^2 + 3*k + 9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

A380746 Number of n-dimensional indecomposable unimodular lattices (or quadratic forms).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A380269 The minimal rank of an n-universal Z-lattice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A379019 Positive integers k such that the simplest cubic field defined by x^3 - k*x^2 - (k+3)*x - 1 is not monogenic.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

A377607 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = 3 is solvable over the integers.

Views

A383068 The set of all integers k >= -1 with the property that there exist integers X and Y such that XY(X+Y) is nonzero and X^3 - kX^2Y - (k+3)XY^2 - Y^3 is a divisor of k^2 + 3*k + 9.

A379019 Positive integers k such that the simplest cubic field defined by x^3 - kx^2 - (k+3)x - 1 is not monogenic.