Seiichi Azuma - cp's OEIS frontend

A383048 Squarefree d such that x^3+y^3=z^3 has nontrivial solution in Q(sqrt(-d)).

2, 5, 6, 11, 14, 15, 17, 23, 26, 29, 31, 33, 35, 38, 41, 42, 47, 51, 53, 59, 62, 65, 69, 71, 74, 77, 78, 83, 86, 87, 89, 95, 101, 105, 106, 107, 109, 110, 113, 114, 119, 122, 123, 129, 131, 134, 137, 141, 143, 146, 149, 155, 158, 159, 161, 167, 170, 173, 174, 177, 179

Offset: 1

Seiichi Azuma, Apr 14 2025

nonn

Equivalent condition is that the elliptic curve dY^2=X^3+432 has positive rank.
If d == 1 (mod 3) appears in this sequence, 3 divides the class number of Q(sqrt(-d)).
For d not divisible by 3, d appears in this sequence if and only if 3d appears in A383047, and 3d appears in this sequence if and only if d appears in A383047.

			For a(1)=2, (-2+sqrt(-2))^3+(-2-sqrt(-2))^3=2^3

M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields, Int. J. Number Theory 9 (2013), no. 6, 1579-1591.

Seiichi Azuma, Table of n, a(n) for n = 1..168
M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields.

Cf. A383047.

for(n=2,500,if(vecmax(factor(n)[,2])>= 2,next); r=ellrank(ellinit([0,0,0,0,432*n^3])); if(r[2]>0,print1(n, ", "); if(r[1]==0,print("uncertain!"))))

a(50) corrected by David Radcliffe, Aug 01 2025

A383047 Squarefree d such that x^3+y^3=z^3 has non-trivial solution in Q(sqrt(d)).

Original entry on oeis.org

2, 5, 6, 11, 14, 15, 17, 23, 26, 29, 33, 35, 38, 41, 42, 43, 47, 51, 53, 58, 59, 62, 65, 69, 71, 74, 77, 78, 82, 83, 85, 86, 87, 89, 93, 95, 101, 105, 106, 107, 109, 110, 113, 114, 119, 122, 123, 131, 134, 137, 141, 142, 143, 146, 149, 155, 158, 159, 161, 167, 170

Offset: 1

Seiichi Azuma, Apr 14 2025

nonn

Equivalent condition is that the elliptic curve dY^2=X^3-432 has positive rank.
Under the Birch and Swinnerton-Dyer conjecture, d not divisible by 3 appears in this sequence if and only if x^2 + y^2 + 7z^2 + xz = d and x^2 + 2y^2 + 4z^2 + xy + yz = d have equal numbers of integral solutions, and d divisible by 3 appears in this sequence if and only if x^2 + 3y^2 + 27z^2 = d/3 and 3x^2 + 4y^2 + 7z^2 - 2yz = d/3 have equal numbers of integral solutions.
For d not divisible by 3, d appears in this sequence if and only if 3d appears in A383048, and 3d appears in this sequence if and only if d appears in A383048.

			For a(1)=2, (18+17*sqrt(2))^3+(18-17*sqrt(2))^3=42^3.

M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields, Int. J. Number Theory 9 (2013), no. 6, 1579-1591.

Seiichi Azuma, Table of n, a(n) for n = 1..160
M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields.

Cf. A383048.

for(n=2,500,if(vecmax(factor(n)[,2])>= 2,next); r=ellrank(ellinit([0,0,0,0,-432*n^3])); if(r[2]>0, print1(n, ", "); if(r[1]==0,print("uncertain!"))))

A355920 Largest prime number p such that x^n + y^n mod p does not take all values on Z/pZ.

Original entry on oeis.org

7, 29, 61, 223, 127, 761, 307, 911, 617, 1741, 1171, 2927, 3181, 2593, 1667, 4519, 2927, 10781, 5167, 6491, 8419, 7177, 5501, 7307, 9829, 11117, 12703, 20011, 10789, 13249, 18217, 22271, 14771, 29629, 13691, 18773, 22543, 21601, 19927, 46411, 18749, 32957

Offset: 3

Seiichi Azuma, Jul 21 2022

nonn

As discussed in the link below, the Hasse-Weil bound assures that x^n + y^n == k (mod p) always has a solution when p - 1 - 2*g*sqrt(p) > n, where g = (n-1)*(n-2)/2 is the genus of the Fermat curve. For example, p > n^4 is large enough to satisfy this condition, so we only need to check finitely many p below n^4.

Conjecture: a(n) == 1 (mod n). - Jason Yuen, May 18 2024

			a(3) = 7 because x^3 + y^3 == 3 (mod 7) does not have a solution, but x^3 + y^3 == k (mod p) always has a solution when p is a prime greater than 7.

Jason Yuen, Table of n, a(n) for n = 3..63
MathOverflow, Does the expression x^4+y^4 take on all values in Z/pZ?
Seiichi Azuma, Python code for calculating a(n) (Google Colab)

Cf. A289559.

import sympy
def a(n):
    g = (n - 1) * (n - 2) / 2
    plist = list(sympy.primerange(2, n ** 4))
    plist.reverse()
    for p in plist:
        # equivalent to p-1-2*g*p**0.5 > n:
        if (p - 1 - n) ** 2 > 4 * g * g * p:
            continue
        solution = [False] * p
        r = sympy.primitive_root(p)
        rn = r ** n % p  # generator for subgroup {x^n}
        d = sympy.n_order(rn, p)  # size of subgroup {x^n}
        nth_power_list = []
        xn = 1
        for k in range(d):
            xn = xn * rn % p
            nth_power_list.append(xn)
            for yn in nth_power_list:
                solution[(xn + yn) % p] = True
        for yn in nth_power_list:  # consider the case x=0
            solution[yn] = True
        solution[0] = True
        if False in solution:
            return p
    return -1
print([a(n) for n in range(3, 18)])

a(13)-a(25) from Jinyuan Wang, Jul 22 2022
a(26)-a(27) from Chai Wah Wu, Sep 13 2022
a(28)-a(33) from Chai Wah Wu, Sep 14 2022
a(34)-a(40) from Robin Visser, Dec 09 2023
a(41)-a(49) from Robin Visser, Mar 18 2024

A247982 Number of inequivalent expressions involving n operands, ignoring sign.

Original entry on oeis.org

1, 5, 47, 733, 15907, 443825, 15129599, 609432421, 28322451979, 1491650309705, 87799033276583, 5711697248522077, 406948643125302163, 31515118552199419745, 2635823964117955940111, 236779241276954626064149, 22736883802469421935493211

Offset: 1

Seiichi Azuma, Sep 28 2014

nonn

Similar to A140606, but for example, (a-b)/(c-d) and (b-a)/(c-d) are also regarded as the same.

Muniru A Asiru, Table of n, a(n) for n = 1..100
Seiichi Azuma, Expressions with n variables

Cf. A140606, A182173.

a(n) = A182173(n)/2. - Zhujun Zhang, Aug 11 2018

a(7)-a(17) from Muniru A Asiru, Aug 13 2018

A237748 Number of different ways to color the vertices of an n-dimensional hypercube using at most 2 colors.

Original entry on oeis.org

4, 6, 23, 496, 2275974, 800648638402240, 1054942853799126580390222487977120, 22436153203535039105819651040959324360753617244078062654624560815030272

Offset: 1

Seiichi Azuma, Feb 12 2014

nonn

Two colorings are regarded as the same if they are conjugated by a permutation of the vertices caused by a rotation of the hypercube.

Regarding mirrored coloring also as the same, the number of ways appears to be given by A000616. - Seiichi Azuma, Feb 14 2014

			For n=2, there are 6 patterns to color the vertices of the square:
..1..1....1..1....1..1....1..0....1..0....0..0
..1..1....1..0....0..0....0..1....0..0....0..0
For example, the next pattern is regarded as the same with the 3rd one above:
..1..0
..1..0

Seiichi Azuma, Number of different ways to color vertices of a n-dimensional hypercube using at most K colors

Cf. A000543.

Column 2 of A325012.

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User: Seiichi Azuma

A383048 Squarefree d such that x^3+y^3=z^3 has nontrivial solution in Q(sqrt(-d)).

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A383047 Squarefree d such that x^3+y^3=z^3 has non-trivial solution in Q(sqrt(d)).

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A355920 Largest prime number p such that x^n + y^n mod p does not take all values on Z/pZ.

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A247982 Number of inequivalent expressions involving n operands, ignoring sign.

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Extensions

A237748 Number of different ways to color the vertices of an n-dimensional hypercube using at most 2 colors.

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Author

Keywords

Comments

Examples

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