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A330938
Numbers that cannot be written as the sum of four proper powers. A proper power is an integer of the form a^b where a,b are integers greater than or equal to 2.
0
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 22, 23, 27
OFFSET
1,2
COMMENTS
There is a proof by Schinzel and Sierpinski that if n >= 33^17 + 12, then n can be written as a sum of four proper powers. Paul Pollack and Enrique Treviño improved that result to find the complete list.
REFERENCES
Paul Pollack & Enrique Treviño, Sums of Proper Powers, The American Mathematical Monthly, 128 (2021), no. 1, p. 40.
A. Schinzel and W. Sierpinski, Sur les puissances propres, Bull. Soc. Roy. Sci. Liege, 34 (1965), pp. 550-554.
EXAMPLE
The first few missing terms are
16 = 2^2 + 2^2 + 2^2 + 2^2,
20 = 2^2 + 2^2 + 2^2 + 2^3,
21 = 2^2 + 2^2 + 2^2 + 3^2,
24 = 2^2 + 2^2 + 2^3 + 2^3,
25 = 2^2 + 2^2 + 2^3 + 3^2,
26 = 2^2 + 2^2 + 3^2 + 3^2,
28 = 2^2 + 2^3 + 2^3 + 2^3.
CROSSREFS
Cf. A001597.
Sequence in context: A123093 A191932 A044920 * A347527 A022766 A249611
KEYWORD
nonn,fini,full
AUTHOR
Enrique Treviño, Jan 03 2020
STATUS
approved