A392342 Numbers that are not the sum of at most four cubefull numbers.

5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 31, 38, 39, 46, 47, 53, 58, 69, 77, 79, 85, 95, 101, 103, 111, 175, 196, 212, 228, 231, 247, 327, 444, 458, 490, 606, 662, 860, 975, 1167, 1470, 1821, 1967, 2039

Offset

1,1

Comments

Erdős and Ivić conjectured every sufficiently large integer is the sum of at most r+1 many r-full numbers, which would imply this sequence is finite. Heath-Brown has proved the conjecture for r=2.

The last known term is a(45) = 2039. There are no other terms < 84000.

There are no other terms < 10^9. See Python program in links. - David Cleaver, Feb 17 2026

References

D. R. Heath-Brown, "Ternary Quadratic Forms and Sums of Three Square-Full Numbers." In Séminaire de Théorie des Nombres, Paris 1986-87 (Ed. C. Goldstein). Boston, MA: Birkhauser, pp. 137-163, 1988.

Links

Table of n, a(n) for n=1..45.

Thomas Bloom, Problem #1107, Erdős Problems.

David Cleaver, Python program.

Example

Smallest cubefull numbers are 1, 8, 16, 27, 32, 64... so no four of them add to 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23 or 31.

Mathematica

n=41000;
t=Join[{0, 1}, Select[Range[2, n], Min[Table[# [[2]], {1}] & /@ FactorInteger[#]] > 2&]];
Complement[Range[n], Flatten[Outer[Plus, t, t, t, t]]]

Crossrefs

Cf. A056828 (r=2), A392343 (r=4).

Cf. A036966 (cubefull numbers).

Sequence in context: A037360 A120521 A022566 * A047320 A327301 A247522

Adjacent sequences: A392338 A392339 A392340 * A392343 A392344 A392345

Keyword

nonn,more

Author

Elijah Beregovsky, Jan 07 2026

Status

approved