Displaying 1-10 of 31 results found.
Numbers that are the sum of 4 distinct nonzero squares in 7 or more ways.
+0
2
190, 198, 210, 222, 231, 238, 246, 255, 270, 282, 286, 290, 294, 302, 303, 306, 310, 315, 318, 326, 330, 334, 335, 338, 342, 345, 350, 351, 354, 357, 358, 363, 366, 369, 370, 374, 375, 378, 381, 382, 385, 386, 387, 390, 393, 394, 398, 399, 402, 405, 406, 407, 410, 411
MAPLE
N:= 1000: # for terms <= N
G:= mul(1+x^(i^2)*y, i=1..floor(sqrt(N))):
G4:= series(coeff(G, y, 4), x, N+1):
A:= select(t -> coeff(G4, x, t) >= 7, [$1..N]); # Robert Israel, Nov 19 2023
MATHEMATICA
With[{nn=25}, Select[Select[Tally[Total/@Subsets[Range[nn]^2, {4}]], #[[2]]> 6&][[All, 1]]//Union, #<=(nn^2-14)&]] (* Harvey P. Dale, Jun 21 2021 *)
Numbers that are the sum of 4 distinct nonzero squares in exactly 5 ways.
+0
2
126, 150, 170, 186, 219, 225, 230, 242, 249, 250, 261, 267, 274, 275, 278, 287, 295, 297, 305, 311, 314, 319, 321, 322, 323, 325, 343, 346, 347, 361, 377, 379, 383, 401, 419, 421, 427, 437, 457, 463, 467, 468, 493, 500, 504, 509, 517, 523, 524, 577, 600, 680, 724, 744
MAPLE
N:= 100000: # for terms <= N
G:= mul(1+x^(i^2)*y, i=1..floor(sqrt(N))):
G4:= series(coeff(G, y, 4), x, N+1):
select(t -> coeff(G4, x, t) = 5, [$1..N]): # Robert Israel, Nov 19 2023
Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0<w<x<y<z.
+0
17
30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137
MATHEMATICA
data = Flatten[ DeleteCases[ FindInstance[ w^2 + x^2 + y^2 + z^2 == # && 0 < w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[137], {}], 1]; w^2 + x^2 + y^2 + z^2 /. data (* Ant King, Oct 17 2010 *) Select[Union[Total[#^2]&/@Subsets[Range[10], {4}]], #<=137&] (* Harvey P. Dale, Jul 03 2011 *)
PROG
(Haskell)
a004433 n = a004433_list !! (n-1)
a004433_list = filter (p 4 $ tail a000290_list) [1..] where
p k (q:qs) m = k == 0 && m == 0 ||
q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
(PARI) list(lim)=my(v=List()); lim\=1; for(z=4, sqrtint(lim\4), for(y=3, min(sqrtint((lim-z^2)\3), z-1), for(x=2, min(sqrtint((lim-y^2-z^2)\2), y-1), for(w=1, min(sqrtint(lim-x^2-y^2-z^2), x-1), listput(v, w^2+x^2+y^2+z^2))))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017
Number of partitions of n into at most 4 distinct positive squares.
+0
1
1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 1, 2, 2, 0, 1, 1, 1, 1, 0, 2, 3, 1, 1, 4, 2, 0, 1, 2, 2, 1, 0, 1, 4, 2, 0, 2, 4, 1, 1, 3, 1, 1, 2, 3, 3, 1, 0, 3, 5, 2, 0, 2, 4, 2, 0, 1, 3, 2, 2, 4
a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such number exists.
+0
9
37, 142, 285, 536, 911, 1268, 1909, 2713, 3876, 5179, 6891, 8901, 11190, 14384, 18087, 21697, 27055, 32166, 39111, 46560, 53892, 64412, 73949, 86778, 98202, 113635, 130088, 148051, 167505, 190968, 214955, 240143, 269775, 297615, 331201, 367429, 409179, 451340, 497830
EXAMPLE
For n = 3: 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.
MATHEMATICA
Do[i=1; While[b=PolygonalNumber[n, Range@i++]; !IntegerQ[t=Min[First/@Select[Tally[Select[Total/@Subsets[b, {n}], #<=Max@b&]], Last@#==n&]]]]; Print@t, {n, 3, 10}] (* Giorgos Kalogeropoulos, Dec 30 2021 *)
Number T(n,k) of partitions of n into k distinct nonzero squares; triangle T(n,k), n>=0, 0<=k<= A248509(n), read by rows.
+0
20
1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1
COMMENTS
T(n,k) is defined for n, k >= 0. The triangle contains only the terms with 0 <= k <= A248509(n). T(n,k) = 0 for k > A248509(n).
FORMULA
T(n,k) = [x^n*y^k] Product_{j>=1} (1 + y*x^(j^2)).
Sum_{k>=0} 2^k * T(n,k) = A279360(n). Sum_{k>=0} k * T(n,k) = A281542(n). Sum_{k>=0} (-1)^k * T(n,k) = A276516(n).
EXAMPLE
T(62,3) = 2 is the first term > 1 and counts partitions [49,9,4] and [36,25,1].
Triangle T(n,k) begins:
1;
0, 1;
0;
0;
0, 1;
0, 0, 1;
0;
0;
0;
0, 1;
0, 0, 1;
0;
0;
0, 0, 1;
0, 0, 0, 1;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i^2>n, 0, expand(b(n-i^2, i-1)*x))))
end:
T:= n->(p->seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, isqrt(n))):
seq(T(n), n=0..45);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
b[n, i - 1] + If[i^2 > n, 0, Expand[b[n - i^2, i - 1]*x]]]];
T[n_] := CoefficientList[b[n, Floor@Sqrt[n]], x] /. {} -> {0};
CROSSREFS
Columns k=0-10 give: A000007, A010052 (for n>0), A025441, A025442, A025443, A025444, A340988, A340998, A340999, A341000, A341001.
Number of partitions of n into 10 distinct nonzero squares.
+0
7
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1
CROSSREFS
Cf. A000144, A000290, A010052, A025434, A025441, A025442, A025443, A025444, A045852, A340947, A340988, A340998, A340999, A341000.
Number of partitions of n into 5 distinct nonzero squares.
+0
8
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0
FORMULA
a(n) = [x^n y^5] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019
EXAMPLE
a(111) = 2 via 1 + 4 + 9 + 16 + 81 = 1 + 9 + 16 + 36 + 49. - David A. Corneth, Feb 02 2021
MAPLE
A025444aux := proc(n, m, nmax) local a, m, upn, lv ; if m = 1 then if issqr(n) and nmax^2 >= n and n >= 1 then return 1; else return 0; end if; else a := 0 ; for upn from 1 to nmax do lv := n-upn^2 ; if lv <0 then break; end if; a := a + procname(lv, m-1, upn-1) ; end do: return a; end if; end proc:
A025444 := proc(n) A025444aux(n, 5, n) ; end proc: (End)
CROSSREFS
Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341000, A341001.
Number of partitions of n into 9 distinct nonzero squares.
+0
7
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 2
EXAMPLE
a(381) = 2 via 1 + 4 + 9 + 16 + 36 + 49 + 64 + 81 + 121 = 1 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100. - David A. Corneth, Feb 02 2021
CROSSREFS
Cf. A000290, A008452, A010052, A025433, A025441, A025442, A025443, A025444, A045851, A340946, A340988, A340998, A340999, A341001.
Number of partitions of n into 8 distinct nonzero squares.
+0
6
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 1
CROSSREFS
Cf. A000143, A000290, A010052, A025432, A025441, A025442, A025443, A025444, A045850, A224983, A340915, A340988, A340998, A341000, A341001.
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