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A307417 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.

%I A307417 #18 Jul 09 2019 12:44:46
%S A307417 8,9,16,17,27,28,29,35,43,54,55,62,64,65,72,91,92,99,118,125,126,127,
%T A307417 128,133,134,152,153,189,190,216,217,224,243,244,250,251,280,307,341,
%U A307417 342,343,344,351,370,371,407,432,433,468,469,512,513,514,520,539,559
%N A307417 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.
%C A307417 There are infinitely many such numbers (proof in the second Johnson link).
%H A307417 César Eliud Lozada, <a href="/A307417/b307417.txt">Table of n, a(n) for n = 1..114</a>
%H A307417 Allan Wm. Johnson Jr., <a href="https://cms.math.ca/crux/backfile/Crux_v5n01_Jan.pdf">Crux Mathematicorum</a>, Vol. 5, No. 1, January 1979, problem 407, 16.
%H A307417 Allan Wm. Johnson Jr., <a href="https://cms.math.ca/crux/backfile/Crux_v5n09_Nov.pdf">Crux Mathematicorum</a>, Vol. 5, No. 9, November 1979, solution to problem 407, 273-277.
%e A307417 a(1) = 8 = [2, 0] (base 4) =  2^3 + 0^3
%e A307417 a(2) = 9 = [2, 1] (base 4) =  2^3 + 1^3
%e A307417 a(3) = 16 = [2, 2] (base 7) =  2^3 + 2^3
%e A307417 a(4) = 17 = [1, 2, 2] (base 3) =  1^3 + 2^3 + 2^3
%p A307417 sqn:= []; lis:=[];
%p A307417 for n to 1000 do
%p A307417   b := 2;
%p A307417   while b < n do #needs to be adjusted
%p A307417     q := convert(n, base, b);
%p A307417     s := convert(map(proc (X) options operator, arrow; X^3 end proc, q), `+`);
%p A307417     if evalb(s = n) then
%p A307417       sqn := [op(sqn), n];
%p A307417       lis := [op(lis), [n, b, ListTools[Reverse](q)]];
%p A307417       break
%p A307417     end if;
%p A307417     b := b+1
%p A307417   end do
%p A307417 end do;
%p A307417 lis := lis; #list of decompositions [number, base, conversion]
%p A307417 sqn := sqn; #sequence
%Y A307417 Cf. A005188, A023052, A046197.
%K A307417 nonn,base
%O A307417 1,1
%A A307417 _César Eliud Lozada_, Apr 07 2019