A066213 -id:A066213

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A066214

Nonsquare numbers which are sums of squares of some subset of divisors.

+10
4

20, 30, 80, 90, 120, 126, 130, 150, 180, 195, 210, 252, 264, 270, 272, 280, 294, 300, 315, 320, 330, 336, 350, 360, 378, 390, 396, 414, 420, 450, 468, 480, 500, 504, 520, 525, 540, 600, 630, 650, 660, 690, 693, 696, 700, 720, 750, 756, 780, 792

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Robert Israel and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2667 terms from Israel)

EXAMPLE

20 is in the list since 20 = 2^2 + 4^2 and 2 and 4 are divisors of 20

MAPLE

N:= 10000; # to get entries up to N

filter:= proc(t)

local L;

L:= select(d -> (d^2<t), numtheory[divisors](t));

evalb(coeff(mul(1+x^(d^2), d=L), x, t) <> 0);

end proc;

A066214:= select(filter, [$2..N]); # Robert Israel, Apr 17 2014

MATHEMATICA

filterQ[n_] := If[IntegerQ[Sqrt[n]], False, Module[{L}, L = Select[ Divisors[n], #<n&]; SeriesCoefficient[Product[1+x^(d^2), {d, L}], {x, 0, n}] != 0]];

Select[Range[1000], filterQ] (* Jean-François Alcover, Jun 07 2020, after Maple *)

okQ[k_] := AnyTrue[Subsets[Select[Divisors[k]^2, # <= k&]], Total[#]==k&];

Reap[For[k = 1, k <= 1000, k++, If[!IntegerQ[k^(1/2)] && okQ[k], Sow[k]]]][[2, 1]] (* Jean-François Alcover, May 27 2024 *)

PROG

(PARI) is(n)=if(issquare(n), return(0)); my(d=divisors(n), v=[0], t); d=apply(sqr, select(k->k^2<n, d)); t=vecsum(d); if(t<n, return(0)); forstep(i=#d, 1, -1, v=concat(apply(k->k+d[i], v), v); t-=d[i]; v=Set(select(k->k<=n && k+t>=n, v)); if(setsearch(v, n), return(1))); 0 \\ Charles R Greathouse IV, Aug 28 2016

CROSSREFS

Cf. A066213, A066215, A066216.

KEYWORD

nonn

AUTHOR

Erich Friedman, Dec 17 2001

STATUS

approved

A066215

Numbers which are sums of cubes of some subset of divisors.

+10
4

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; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10286 (first 649 terms from R. J. Mathar)

R. J. Mathar, Examples/decompositions for entries <10000

EXAMPLE

72 is in the list since 72 = 2^3 + 4^3 and 2 and 4 are divisors of 72

MATHEMATICA

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 *)

CROSSREFS

Cf. A000578, A066214, A066213, A066216.

KEYWORD

nonn

AUTHOR

Erich Friedman, Dec 17 2001

STATUS

approved

A066216

Noncube numbers which are sums of cubes of some subset of divisors.

+10
4

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)

OFFSET

1,1

LINKS

Jean-François Alcover, Table of n, a(n) for n = 1..1600

EXAMPLE

72 is in the list since 72 = 2^3 + 4^3 and 2 and 4 are divisors of 72

MATHEMATICA

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 *)

CROSSREFS

Cf. A066214, A066215, A066213.

KEYWORD

nonn

AUTHOR

Erich Friedman, Dec 17 2001

STATUS

approved

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