A385316 Smallest number that is the sum of 3 cubes of primes in exactly n different ways.

24, 185527, 8627527, 999979163, 10588881419

Offset

1,1

Comments

In the SageMath code, B > 326, is sufficient to conclude the correctness of the 5th term (a(5) < P[327]^3) and B > 168 to confirm the first 4 values.

For the 6th term, one needs B > 1001 implying a(6) > 499243435237.

Links

Table of n, a(n) for n=1..5.

Thomas Bloom, Problem 979, Erdős Problems.

Example

a(1) = 24, since 24 = 2^3 + 2^3 + 2^3 (unique way) is the smallest sum of prime cubes.
a(5) = 10588881419, since it is the smallest number which is sum of 3 primecubes in 5 ways, it can be obtained with the following triples: {59, 1669, 1811}, {83, 1567, 1889}, {139, 1427, 1973}, {349, 1091, 2099} and {479, 929, 2131}.

Prog

(SageMath)
B=327
P=Primes()
d={}
for i in P[0:B]:
    d[3*i**3]=1;
for (i, j, k) in Combinations(P[0:B], 3):
    if i**3+j**3+k**3 in d.keys():
        d[i**3+j**3+k**3]+=1
    else:
        d[i**3+j**3+k**3]=1
for (i, j) in Combinations(P[0:B], 2):
    if i**3+2*j**3 in d.keys():
        d[i**3+2*j**3]+=1
    else:
        d[i**3+2*j**3]=1
    if 2*i**3+j**3 in d.keys():
        d[2*i**3+j**3]+=1
    else:
        d[2*i**3+j**3]=1
M=[i for i in d.keys() if d[i]==5]
min(M)

Crossrefs

Cf. A255018, A025418, A025419, A124980, A088919.

Sequence in context: A201495 A258924 A179268 * A056947 A305760 A048057

Adjacent sequences: A385313 A385314 A385315 * A385317 A385318 A385319

Keyword

nonn,hard,more

Author

Stijn Cambie, Sep 20 2025

Status

approved