Displaying 101-110 of 113 results found.
Taxicab numbers (sums of 2 cubes in more than 1 way) which are products of five distinct primes.
+10
1
16387189, 16776487, 17045567, 24767171, 38253878, 39639691, 40183262, 41892515, 44409995, 51278929, 60271939, 73842713, 106496767, 122810129, 129380329, 145908847, 154245637, 156234169, 176427433, 197842337, 243578881, 271688534, 272264167, 292940137, 300694303, 373333697, 389675503, 401947273
COMMENTS
A squarefree subsequence of taxicab numbers.
EXAMPLE
16387189 = 5^3 + 254^3 = 197^3 + 206^3 = 7*13*31*37*157.
38253878 = 87^3 + 335^3 = 173^3 + 321^3 = 2*13*19*211*367.
44409995 = 138^3 + 347^3 = 176^3 + 339^3 = 5*7*97*103*127.
Values of the argument at successive record minima of the function R defined as follows. For any integer x >= 1, let y > x be the smallest integer such that there exist integers x < c < d < y such that x^3 + y^3 = c^3 + d^3. Then R(x) = y/x.
+10
1
1, 2, 8, 9, 10, 17, 30, 42, 51, 135, 156, 285, 792, 1634, 3751, 4026, 6192, 14934, 15768, 16147, 45121, 58230, 61389, 79876, 167757, 177560, 213652, 525537, 917324, 1050787, 2237052, 3954983, 4157802
COMMENTS
For a given integer x, the identity x^3 + (12x)^3 = (9x)^3 + (10x)^3 holds, so R(x) <= 12.
A quadruple x = 2*N^4 - 4*N^3 + 9*N^2 - 8*N +10, y = 2*N^4 + 6*N^2 + N + 9, c = 2*N^4 - 3*N^3 + 12*N^2 - 5*N + 12, d = 2*N^4 - N^3 + 6*N^2 + N + 1 (for integer N) shows that the sequence is infinite.
EXAMPLE
For x = 1, y = 12, 1^3 + 12^3 = 9^3 + 10^3, R(1) = 12. So, a(1) = 1.
For x = 2, y = 16, 2^3 + 16^3 = 9^3 + 15^3, R(2) = 8. So, a(2) = 2.
For x = 3, y = 36, 3^3 + 36^3 = 27^3 + 30^3, R(3) = 12. So, this does not provide a record minimum. The same negative outcome happens for x = 4, x = 5, x = 6, x = 7.
For x = 8, y = 53, 8^3 + 53^3 = 29^3 + 50^3, R(8) = 6.625. So, a(4) = 8.
For n = 8, a(8) = 42, since 42^3 + 69^3 = 56^3 + 61^3, and the ratio R(42) = 69/42 = 1.6428571... is an absolute minimum (eighth successive) for the function R(x) for 1 <= x <= 42.
PROG
(Python)
xm, ym, x, n = 0, 1, 0, 1
while True:
x, y = x+1, x+4
while y*xm < ym*x:
c, d, s = x+1, y-1, x**3+y**3
while c<d:
t=c**3+d**3
if t<s:
c+=1
elif t>s:
d-=1
else:
break
if t==s:
print("a({})={} x={} c={} d={} y={}".format(n, x, x, c, d, y))
xm, ym, n = x, y, n+1
break
y+=1
w such that w^3+x^3+y^3+z^3=0, w>|x|>|y|>|z|, is soluble.
+10
0
6, 9, 12, 16, 19, 20, 25, 27, 28, 29, 34, 39, 40, 41, 44, 46, 51, 53, 54, 55, 58, 60, 67, 69, 70, 71, 72, 75, 76, 80, 81, 82, 84, 85, 87, 88, 89, 90, 93, 94, 96, 97, 98, 99, 102, 103, 105, 108, 109, 110, 111, 113, 115, 116, 120, 121, 122, 123, 126, 127, 129, 132, 134, 137, 139
REFERENCES
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780, see p. 799.
H. W. Richmond, On integers which satisfy ..., Trans. Camb. Phil. Soc., 22 (1920), 389-403, see p. 402.
MATHEMATICA
sol[w_] := Reap[ Do[ If[ GCD[w, x, y, z] == 1 && w > Abs[x] > Abs[y] > Abs[z] && w^3 + x^3 + y^3 + z^3 == 0, Print[{w, x, y, z}]; Sow[{w, x, y, z}]; Break[]], {x, -w+1, -1}, {y, x+1, -1}, {z, y+1, -y-1}]][[2]]; Select[ Range[140], sol[#] =!= {} & ] (* Jean-François Alcover, Feb 24 2012 *)
Numbers of the form x^3+y^5, with x,y>0 and x<>y.
+10
0
9, 28, 33, 59, 65, 96, 126, 157, 217, 244, 248, 251, 307, 344, 368, 375, 459, 513, 544, 586, 730, 755, 761, 972, 1001, 1025, 1032, 1032, 1051, 1149, 1240, 1243, 1332, 1363, 1367, 1536, 1574, 1729, 1753, 1760, 1971, 2024, 2198, 2229, 2355, 2440, 2745, 2752
COMMENTS
Numbers with more than one of these representations are repeated for each of them.
This concerns 1032 = 2^3+4^5 = 10^3+2^5 or 9504 = 12^3+6^5 = 21^3+3^5, for example (see A035046).
EXAMPLE
59=3^3+2^5, 157=5^3+2^5, 513=8^3+1^5, 586=7^3+3^5, ...
MATHEMATICA
lst={}; Do[Do[Do[If[x!=y, a=x^3+y^5; If[a>n, Break[]]; If[a==n, AppendTo[lst, n]]], {y, 5!}], {x, 5!}], {n, 7!}]; lst
CROSSREFS
Cf. A088719, A088677, A088703, A088687, A001235, A024670, A025320, A025319, A025318, A025317, A025316, A025315, A025314, A025313, A024508, A004431, A024507, A155468, A155469, A155470, A155472
1, 3, 9, 11, 33, 99, 121, 363, 1089
COMMENTS
Divisors of 33^2.
The sum of divisors of 1089 is equal to 1729, the Hardy-Ramanujan number: A000203(1089) = A001235(1) = 1729. The aliquot divisors of 1089 are also the powers of 3 written in base 26, see A004668. Also 1089 is widely used in magic tricks because it can be produced from any two three-digit numbers (see Links section).
EXAMPLE
Sigma(1089) = 1 + 3 + 9 + 11 + 33 + 99 + 121 + 363 + 1089 = 1729.
Decimal expansion of the cube root of 1729.03.
+10
0
1, 2, 0, 0, 2, 3, 8, 3, 7, 8, 5, 6, 9, 1, 7, 1, 8, 1, 2, 3, 0, 5, 7, 3, 8, 1, 6, 6, 9, 9, 5, 0, 4, 4, 0, 4, 0, 7, 5, 0, 6, 8, 5, 1, 2, 2, 0, 5, 0, 8, 9, 2, 7, 5, 3, 6, 0, 2, 8, 8, 1, 3, 0, 7, 3, 3, 9, 5, 0, 2, 4, 2, 1, 2, 7, 6, 7, 9, 4, 4, 6, 5, 6, 3, 4, 3, 0, 2, 0, 1, 0, 9, 6, 8, 0, 8, 2, 0, 3, 2, 3, 0, 8, 4, 2
COMMENTS
The problem of extracting this cube root pitted an abacus salesman against Nobel Prize winning physicist Richard Feynman one afternoon in Rio de Janeiro.
An algebraic number of degree 3 and denominator 10; minimal polynomial 100x^3 - 172903. - Charles R Greathouse IV, Apr 20 2016
REFERENCES
Richard Feynman and Ralph Leighton, Surely You're Joking, Mr. Feynman! (Adventures of a Curious Character), chapter "Lucky Numbers," W. W. Norton & Co., NY 1985, pp. 192-198.
Dana Mackenzie, The Universe in Zero Words, The Story of Mathematics as Told Through Equations, Princeton University Press, Princeton and Oxford, 2012, Introduction - The Abacist versus the Algorist, page 13.
EXAMPLE
12.002383785691718123057381669950440407506851220508927536028813073395024212767944...
MATHEMATICA
RealDigits[ 1729030^(1/3), 10, 105][[1]] (* please notice the lack of a decimal point *)
Taxicab palindromes: palindromes in base 10 that are the sum of two positive cubes in more than one way.
+10
0
4607064, 2344344434432, 109514040415901
EXAMPLE
4607064 = 32^3 + 166^3 = 129^3 + 135^3.
2344344434432 = 2772^3 + 13244^3 = 7161^3 + 12551^3.
109514040415901 = 18909^3 + 46838^3 = 20108^3 + 46629^3.
Cubefree taxi-cab numbers that are not squarefree.
+10
0
40033, 443889, 1845649, 2048391, 4342914, 5799339, 26122131, 32973759, 41301953, 45882739, 53226297, 54269091, 65272753, 66763333, 70449093, 84637287, 86316741, 90527229, 91140435, 94100426, 104212017, 127396178, 128966383, 131126303, 131997229
COMMENTS
There are two versions of "taxicab numbers" that are A001235 and A011541. This sequence focuses on the version A001235. This sequence lists cubefree taxi-cab numbers that are divisible by a square greater than 1.
EXAMPLE
Taxi-cab number 40033 is a term because 40033 = 7^2*19*43.
Taxi-cab number 443889 is a term because 443889 = 3^2*31*37*43.
Taxi-cab number 1845649 is a term because 1845649 = 13^2*67*163.
PROG
(PARI) T = thueinit(x^3+1, 1);
isA001235(n) = my(v=thue(T, n)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1;
isA067259(n) = n>3 && vecmax(factor(n)[, 2])==2;
for(n=1, 1e10, if(isA001235(n) && isA067259(n), print1(n, ", ")));
Least number that is the sum of 2 positive cubes in exactly n ways and also the sum of 2 positive squares in exactly n ways.
+10
0
COMMENTS
4624776 is the first term of A272701. A011541(k) is not the sum of two nonzero squares for 2 <= k <= 6.
If it exists, what is the a(3)?
EXAMPLE
a(1) = 2 because 2 = 1^3 + 1^3 = 1^2 + 1^2.
a(2) = 4624776 because 4624776 = 51^3 + 165^3 = 72^3 + 162^3 = 1026^2 + 1890^2 = 1350^2 + 1674^2.
Values of A272701 that are the sum of a positive square and a positive cube in more than one way.
+10
0
36998208, 449519625, 2367885312, 8016025680, 9563569561, 14753560033, 26971693632, 28769256000, 61358997609, 151544659968, 225128651328, 278450575201, 282429583137, 310289733000, 310289733000, 327699806625, 498700534033, 513025643520, 578097000000
COMMENTS
Taxi-cab numbers ( A001235) that are the sum of two nonzero squares in more than one way and also the sum of a positive square and a positive cube in more than one way. A001235(293) = 6^3* A001235(16) = 6^3*171288 = 36998208 is the least number with this property.
14753560033 = 1453*2677*3793 is the first term that is in A272935. Obviously, in this sequence there are perfect powers infinitely many times.
EXAMPLE
36998208 is a term because 36998208 = 102^3 + 330^3 = 144^3 + 324^3 = 1728^2 + 324^3 = 5832^2 + 144^3 = 648^2 + 6048^2 = 1728^2 + 5832^2.
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