Search: a066214 -id:a066214
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A066213
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Numbers which are sums of squares of some subset of divisors.
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+10
4
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1, 4, 9, 16, 20, 25, 30, 36, 49, 64, 80, 81, 90, 100, 120, 121, 126, 130, 144, 150, 169, 180, 195, 196, 210, 225, 252, 256, 264, 270, 272, 280, 289, 294, 300, 315, 320, 324, 330, 336, 350, 360, 361, 378, 390, 396, 400, 414, 420, 441, 450, 468, 480, 484, 500
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listen;
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internal format)
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OFFSET
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1,2
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COMMENTS
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If m is in the sequence then so is m*k^2 for k >= 1. - David A. Corneth, Jan 22 2024
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LINKS
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EXAMPLE
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20 is in the list since 20 = 2^2 + 4^2 and 2 and 4 are divisors of 20
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MAPLE
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isA066213 := proc(n)
local S, els;
S:=subsets(numtheory[divisors](n));
while not S[finished] do
els:=S[nextvalue]() ;
if add(d^2, d=els) = n then
return true ;
end if ;
end do;
false
end proc:
for n from 1 do
if isA066213(n) then
print(n) ;
end if;
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MATHEMATICA
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okQ[k_] := AnyTrue[Subsets[Select[Divisors[k]^2, # <= k&]], Total[#]==k&];
Reap[For[k = 1, k <= 5000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, May 27 2024 *)
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PROG
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(PARI) \\ See PARI link
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A066215
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Numbers which are sums of cubes of some subset of divisors.
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+10
4
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1, 8, 27, 36, 64, 72, 125, 216, 252, 288, 343, 378, 512, 520, 576, 584, 729, 738, 756, 792, 828, 855, 954, 972, 1000, 1044, 1331, 1350, 1440, 1520, 1540, 1728, 1764, 1800, 1890, 1944, 1980, 2016, 2070, 2160, 2197, 2304, 2352, 2376, 2400, 2484, 2520, 2548
(list;
graph;
refs;
listen;
history;
text;
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OFFSET
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1,2
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COMMENTS
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There are cubes that have not a single, trivial representation but more than one. These start with 27000 =+8^3+9^3+10^3+12^3+15^3+18^3+24^3 =+1^3+2^3+4^3+6^3+8^3+10^3+15^3+20^3+24^3 = +2^3+4^3+9^3+10^3+15^3+20^3+24^3 =+1^3+2^3+4^3+12^3+15^3+20^3+24^3 =+2^3+3^3+4^3+5^3+6^3+12^3+15^3+18^3+25^3 =+1^3+4^3+5^3+6^3+8^3+9^3+12^3+20^3+25^3 =+15^3+20^3+25^3 = +1^3+3^3+6^3+8^3+9^3+18^3+27^3 =+30^3 and 46656 =+1^3+2^3+3^3+6^3+8^3+9^3+12^3+16^3+18^3+24^3+27^3 =+4^3+24^3+32^3 =+36^3 and 74088 =+2^3+6^3+7^3+8^3+9^3+12^3+18^3+21^3+24^3+27^3+28^3 =+4^3+6^3+8^3+14^3+18^3+21^3+24^3+27^3+28^3 =+42^3. - R. J. Mathar, Jan 21 2024
If m is in the sequence then so is m*k^3 for k >= 1. - David A. Corneth, Jan 21 2024
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LINKS
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EXAMPLE
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72 is in the list since 72 = 2^3 + 4^3 and 2 and 4 are divisors of 72
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MATHEMATICA
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okQ[k_] := AnyTrue[Subsets[Select[Divisors[k]^3, # <= k&]], Total[#]==k&];
Reap[For[k = 1, k <= 10000, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, May 27 2024 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A066216
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Noncube numbers which are sums of cubes of some subset of divisors.
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+10
4
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36, 72, 252, 288, 378, 520, 576, 584, 738, 756, 792, 828, 855, 954, 972, 1044, 1350, 1440, 1520, 1540, 1764, 1800, 1890, 1944, 1980, 2016, 2070, 2160, 2304, 2352, 2376, 2400, 2484, 2520, 2548, 2556, 2700, 2772, 2808, 2820, 2870, 2880, 3024, 3220, 3240
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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LINKS
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EXAMPLE
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72 is in the list since 72 = 2^3 + 4^3 and 2 and 4 are divisors of 72
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MATHEMATICA
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okQ[k_] := AnyTrue[Subsets[Select[Divisors[k]^3, # <= k&]], Total[#]==k&];
Reap[For[k = 1, k <= 100000, k++, If[!IntegerQ[k^(1/3)] && okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, May 27 2024 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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