A392343 Numbers that are not the sum of at most five 4-full numbers.

6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 101, 102, 103, 104, 105

Offset

1,1

Comments

Erdős and Ivić conjectured every sufficiently large integer 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.

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..70.

Thomas Bloom, Problem #1107, Erdős Problems.

Mathematica

n=31000;
t=Join[{0, 1}, Select[Range[2, n], Min[Table[# [[2]], {1}] & /@ FactorInteger[#]] > 3&]];
Complement[Range[n], Flatten[Outer[Plus, t, t, t, t, t]]]

Crossrefs

Cf. A056828 (r=2), A392342 (r=3).

Cf. A036967 (4-full numbers).

Sequence in context: A028253 A020715 A380486 * A182307 A023384 A053407

Adjacent sequences: A392339 A392340 A392342 * A392344 A392345 A392346

Keyword

nonn

Author

Elijah Beregovsky, Jan 07 2026

Status

approved