A392342 - OEIS

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

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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.

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

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.

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).