A380448 Least primes which can be represented as the sum of distinct positive cubes in exactly n ways, or 0 if no such prime exists.

73, 757, 953, 2521, 1801, 3257, 2953, 4013, 4139, 4789, 5347, 4481, 5669, 4663, 6427, 6659, 5867, 6301, 6841, 7867, 7687, 7741, 10169, 7057, 7723, 7561, 9631, 8443, 8191, 8387, 9883, 10079, 10313, 10891, 10729, 10009, 9109, 10711, 9829, 11161, 10457, 12547, 11699, 10513, 10333, 11159, 13007

Offset

1,1

Comments

From David A. Corneth, Jun 22 2025: (Start)

a(234) = 0. We have if a(234) > 0 then a(234) > 10^6.

Proof:

Let Q(k, u) be the number of ways to write k as a sum of distinct cubes c where c <= u^3. Then for all m we have Q(k, m+1) >= Q(k, m).

for 39031 <= k <= 633371 we have Q(k, 40) >= 235.

Those are 633371 - 39031 + 1 = 594341 > 41^3 consecutive values for k.

Therefore Q(k, 41) >= 235 for 39031 <= k <= 633371 + 26^3.

We have n^3 + (n+1)^3 > (n+2)^3 for n >= 6. So via induction we have Q(k, u) > 234 for some u for any k >= 39031 which completes the proof.

Similarly Q(k, 40) > 10000 for 100000 <= k <= 500000 which is more than enough to confirm zero's found in the first 10000 terms in a search up to 10^6. (End)

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 233 terms from Zhining Yang)

Example

a(4) = 2521 because 2521 = 4^3 + 9^3 + 12^3 = 1^3 + 4^3 + 5^3 + 10^3 + 11^3= 1^3 +4^3+ 6^3 + 8^3 + 12^3 =4^3 + 6^3 + 8^3 + 9^3 + 10^3 and 2521 is the least prime that can be written as the sum of distinct positive cubes in 4 different ways.

Mathematica

s = CoefficientList[Series[Product[(1 + x^(r^3)), {r, 20}], {x, 0, 8000}], x];
Table[SelectFirst[Flatten@Position[s, k] - 1, PrimeQ], {k, 20}]

Crossrefs

Cf. A003997, A275154, A385094.

Sequence in context: A133753 A177514 A137835 * A381713 A063784 A066101

Adjacent sequences: A380445 A380446 A380447 * A380449 A380450 A380451

Keyword

nonn

Author

Zhining Yang, Jun 21 2025

Status

approved