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A307417
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Numbers that can be expressed in a base in such a way that the sum of cubes of their digits in this base equals the original number.
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1
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8, 9, 16, 17, 27, 28, 29, 35, 43, 54, 55, 62, 64, 65, 72, 91, 92, 99, 118, 125, 126, 127, 128, 133, 134, 152, 153, 189, 190, 216, 217, 224, 243, 244, 250, 251, 280, 307, 341, 342, 343, 344, 351, 370, 371, 407, 432, 433, 468, 469, 512, 513, 514, 520, 539, 559
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OFFSET
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1,1
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COMMENTS
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There are infinitely many such numbers (proof in the second Johnson link).
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LINKS
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Allan Wm. Johnson Jr., Crux Mathematicorum, Vol. 5, No. 9, November 1979, solution to problem 407, 273-277.
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EXAMPLE
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a(1) = 8 = [2, 0] (base 4) = 2^3 + 0^3
a(2) = 9 = [2, 1] (base 4) = 2^3 + 1^3
a(3) = 16 = [2, 2] (base 7) = 2^3 + 2^3
a(4) = 17 = [1, 2, 2] (base 3) = 1^3 + 2^3 + 2^3
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MAPLE
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sqn:= []; lis:=[];
for n to 1000 do
b := 2;
while b < n do #needs to be adjusted
q := convert(n, base, b);
s := convert(map(proc (X) options operator, arrow; X^3 end proc, q), `+`);
if evalb(s = n) then
sqn := [op(sqn), n];
lis := [op(lis), [n, b, ListTools[Reverse](q)]];
break
end if;
b := b+1
end do
end do;
lis := lis; #list of decompositions [number, base, conversion]
sqn := sqn; #sequence
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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