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A346071 a(n) is the smallest number m such that m^3 = x^3 + y^3 + z^3, x > y > z > 0, has at least n different solutions. 1
6, 18, 54, 87, 108, 174, 174, 324, 324, 324, 492, 492, 492, 984, 984, 1296, 1296, 1296, 1440, 1440, 2592, 2592, 2592, 2592, 3960, 3960, 3960, 3960, 4320, 4320, 4320, 5760, 5940, 5940, 5940, 5940, 5940, 5940, 8640, 9900, 9900, 9900, 11880, 11880, 11880, 11880, 11880 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) is the smallest number for which there are at least n sets of positive integers (b_i, c_i, d_i) i=1..n which satisfy the equation a(n)^3 = b_i^3 + c_i^3 + d_i^3.

This sequence is related to Euler's sum of powers conjecture. In particular to the case k=3, a(n) is the smallest number that has at least n different solutions to the equation.

The sequences of numbers whose cubes can be expressed as the sum of 3 positive cubes in at least n ways for n = 1, 2, 3, ... form a family of related sequences. This sequence is the sequence of first terms in that family of sequences.

The first of this family is A023042.

LINKS

Table of n, a(n) for n=1..47.

Wikipedia, Euler's sum of powers conjecture

EXAMPLE

a(1) = 6 because 6^3 = 5^3 + 4^3 + 3^3; 6 = a(1) = A023042(1).

a(2) = 18 because 18^3 = 15^3 + 12^3 + 9^3 = 16^3 + 12^3 + 2^3.

a(3) = 54 because 54^3 = 45^3 + 36^3 + 27^3 = 48^3 + 36^3 + 6^3 = 53^3 + 19^3 + 12^3.

PROG

(Python)

import numpy as np

def residual(a, b, c, d, exp=3):

    return a**exp-b**exp-c**exp-d**exp

def test(max_n, k=3):

    ans=dict()

    for a in range(max_n):

        #print(a)

        for b in range(int(np.ceil((a**k/3)**(1/k))), a):

            n3=a**k-b**k

            for c in range(int(np.ceil((n3/2)**(1/k))), b):

                m3=n3-c**k

                if m3<0:

                    break;

                l=int(np.ceil((m3)**(1/k)))

                options=[l, l-1]

                for d in options:

                    res=residual(a, b, c, d, exp=k)

                    if res==0:

                        if a in ans.keys():

                            ans[a].append((a, b, c, d))

                        else:

                            ans[a]=[(a, b, c, d)]

                        #print("found:", (a, b, c, d))

                        break

                    else:

                        #print("tested: {0}, residual: {1}".format((a, b, c, d), res))

                        if res>0:

                            break

    return ans

def serie(N):

    result=test(N)

    results_by_number_of_answers=[]

    results_by_number_of_answers.append(result)

    temp=dict()

    for k in result.keys():

        if len(result[k])>=2:

            temp[k]=result[k]

    results_by_number_of_answers.append(temp)

    i=3

    while len(temp)>0:

        temp=dict()

        for k in results_by_number_of_answers[-1].keys():

            if len(results_by_number_of_answers[-1][k])>=i:

                temp[k]=result[k]

        if len(temp)>0:

            results_by_number_of_answers.append(temp)

        i+=1

    return [next(iter(a)) for a in results_by_number_of_answers]

#Get the elements of the serie up until A_n>1000

A=serie(1000)

print(A)

(Python)

from itertools import combinations

from collections import Counter

from sympy import integer_nthroot

def icbrt(n): return integer_nthroot(n, 3)[0]

def aupto(mmax):

    cbs = [i**3 for i in range(mmax+1)]

    cbsset = set(cbs)

    c = Counter(sum(c) for c in combinations(cbs, 3) if sum(c) in cbsset)

    nmax = max(c.values())

    return [min(icbrt(s) for s in c if c[s] >= n) for n in range(1, nmax+1)]

print(aupto(500)) # Michael S. Branicky, Sep 04 2021

CROSSREFS

Cf. A023042, A025418, A346137, A316359.

Sequence in context: A318484 A079843 A290582 * A076941 A006779 A003208

Adjacent sequences:  A346068 A346069 A346070 * A346072 A346073 A346074

KEYWORD

nonn,more

AUTHOR

Sebastian Magee, Jul 30 2021

EXTENSIONS

a(16)-a(31) from Jinyuan Wang, Aug 02 2021

More terms from David A. Corneth, Sep 04 2021

STATUS

approved

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Last modified December 2 11:59 EST 2021. Contains 349440 sequences. (Running on oeis4.)