Search: a175365 -id:a175365
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1, 7, 19, 27, 27, 27, 27, 27, 33, 57, 81, 81, 81, 81, 81, 81, 93, 117, 117, 117, 117, 117, 117, 117, 125, 125, 125, 131, 155, 179, 179, 179, 179, 179, 179, 203, 251, 251, 251, 251, 251, 251, 251, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 287, 311
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OFFSET
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0,2
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COMMENTS
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Number of integer triples (x,y,z) satisfying |x|^3+|y|^3+|z|^3 <= n, -n <= x,y,z <= n. A variant of A117609 with cubes instead of squares.
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LINKS
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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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A175368
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Number of integer 4-tuples (x,y,z,u) satisfying |x|^3 + |y|^3 + |z|^3 + |u|^3 = n, -n <= x,y,z,u <= n.
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+10
2
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1, 8, 24, 32, 16, 0, 0, 0, 8, 48, 96, 64, 0, 0, 0, 0, 24, 96, 96, 0, 0, 0, 0, 0, 32, 64, 0, 8, 48, 96, 64, 0, 16, 0, 0, 48, 192, 192, 0, 0, 0, 0, 0, 96, 192, 0, 0, 0, 0, 0, 0, 64, 0, 0, 24, 96, 96, 0, 0, 0, 0, 0, 96, 192, 8, 48, 96, 64, 0, 0, 96, 0, 48, 192, 192, 0, 0, 0, 0, 0, 96, 224, 64, 0
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OFFSET
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0,2
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COMMENTS
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A variant of A000118 with cubes instead of squares.
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LINKS
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FORMULA
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Conjectured g.f.: (1 + 2*Sum_{j>=1} x^(j^3))^4.
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EXAMPLE
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a(1) = 8 counts (x,y,z,u) = (-1,0,0,0), (0,-1,0,0), (0,0,-1,0), (0,0,0,-1) and 4 more tuples with -1 replaced by +1.
a(2) = 24 counts (x,y,z,u) = (-1,-1,0,0), (-1,0,-1,0), (-1,0,0,-1), (-1,0,0,1) etc, all variants where two of the 4 values are zero and the other two +1 or -1.
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PROG
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(PARI) a(n, k=4) = if(n==0, return(1)); if(k <= 0, return(0)); if(k == 1, return(ispower(n, 3))); my(count = 0); for(v = 0, sqrtnint(n, 3), count += (2 - (v == 0))*if(k > 2, a(n - v^3, k-1), if(ispower(n - v^3, 3), 2 - (n - v^3 == 0), 0))); count; \\ Daniel Suteu, Aug 15 2021
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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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A338932
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Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.
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+10
1
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1, 2, 128, 729, 1458, 4096, 65536, 93312, 2985984, 3906250, 16777216, 28697814, 33554432, 47775744, 80707214, 244140625, 250000000, 387420489, 1836660096, 2847656250, 4715895382, 5165261696, 12230590464, 13841287201, 17179869184, 21208998746, 24461180928
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OFFSET
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1,2
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COMMENTS
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The data are derived from the following formula:
(a^3 - 6*t^3)^3 + (a^3 + 6*t^3)^3 + (-6*a*t^2)^3 = 2*a^9;
(4*a^3 - 3*t^3)^3 + (4*a^3 + 3*t^3)^3 + (-6*a*t^2)^3 = 128*a^9 = 2*4^3*a^9;
(9*a^3 - 2*t^3)^3 + (9*a^3 + 2*t^3)^3 + (-6*a*t^2)^3 = 1458*a^9 = 2*9^3*a^9;
(36*a^3 - t^3)^3 + (36*a^3 + t^3)^3 + (-6*a*t^2)^3 = 93312*a^9 = 2*36^3*a^9;
((3*a^3)*t - 9*t^4)^3 + (9*t^4)^3 + (a^4 - 9*a*t^3)^3 = a^12;
((9*a^3)*t - t^4)^3 + (t^4)^3 + (9*a^4 - 3*a*t^3)^3 = 729*a^12 = 9^3*a^12.
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, D5.
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LINKS
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EXAMPLE
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128 is a term, because (4 - 3*(2*n - 1)^3, 4 + 3*(2*n - 1)^3, -3*(2*n - 1)^2) is a nontrivial primitive parametric solution of x^3 + y^3 + z^3 = 128.
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MATHEMATICA
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t1 = 2*{1, 5, 7, 11, 13}^9;
t2 = 128*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*{1, 3, 5, 7, 9}^9;
t4 = 93312*{1, 2, 3, 4, 5}^9;
t5 = {1, 2, 4, 5, 7}^12;
t6 = 729*{1, 2, 3, 4, 5}^12;
Take[Union[t1, t2, t3, t4, t5, t6], 27]
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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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A338933
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Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.
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+10
1
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2, 16, 128, 1024, 1458, 8192, 11664, 31250, 65536, 93312, 235298, 524288, 746496, 1062882, 2000000, 3543122, 3906250, 5971968, 9653618, 15059072, 22781250, 28697814, 33554432, 47775744, 48275138, 68024448, 80707214, 94091762, 128000000, 171532242, 226759808
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OFFSET
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1,1
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COMMENTS
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The data are derived from the following formula:
(a^2 - a*t - t^2)^3 + (a^2 + a*t - t^2)^3 + 2*(t^2)^3 = 2*a^6
(a^3 - 3*t^3)^3 + (a^3 + 3*t^3) + 2*(-3*a*t^2)^3 = 2*a^9;
(9*a^3 - t^3)^3 + (9*a^3 + t^3)^3 + 2*(-3*a*t^2)^3 = 1458*a^9;
(6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;
(6*a^3*t - 9*t^4)^3 + (9*t^4)^3 + 2*(2*a^4 - 9*a*t^3)^3 = 16*a^12 = 2*2^3*a^12;
(18*a^3*t - 8*t^4)^3 + (8*t^4)^3 + 2*(9*a^4 - 12*a*t^3)^3 = 1458*a^12 = 2*9^3*a^12;
(18*a^3*t - t^4)^3 + (t^4)^3 + 2*(18*a^4 - 3*a*t^3)^3 = 11664*a^12 = 2*18^3*a^12.
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, D5.
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LINKS
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EXAMPLE
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16 is a term, because when t is an integer, (6*(2*t + 1) - 9*(2*t + 1)^4, 9*(2*t + 1)^4, 2 - 9*(2*t + 1)^3) is a nontrivial primitive parametric solution of x^3 + y^3 + 2*z^3 = 16.
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MATHEMATICA
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t1 = 2*Range[23]^6;
t2 = 2*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*Range[4]^9;
t4 = 2*{1, 5}^12;
t5 = 16*{1, 2, 4}^12;
t6 = 1458*{1, 3}^12;
t7 = 11664*{1, 2, 3}^12;
Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]
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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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Missing terms 1024 and 746496 added by XU Pingya, Mar 14 2022
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STATUS
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approved
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