A360427
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.
Original entry on oeis.org
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
Offset: 1
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.
-
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 cs:
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
# Bert Dobbelaere, Mar 18 2023
A268757
Unhappycab numbers: the smallest unhappy number that is the sum of two cubes of unhappy numbers in n different ways.
Original entry on oeis.org
16, 4104, 119824488, 74213505639000, 6123582409620312000
Offset: 1
a(1) = 16 = 2^3+2^3;
a(2) = 4104 = 2^3+16^3 = 9^3+15^3;
a(3) = 119824488 = 11^3+493^3 = 90^3+492^3 = 346^3+428^3;
a(4) = 74213505639000 = 5895^3+41985^3 = 20392^3+40358^3 = 20880^3+40230^3 = 32790^3+33900^3;
a(5) = 6123582409620312000 = 193935^3+1828785^3 = 539195^3+1813765^3 = 1026460^3+1714760^3 = 1107120^3+1682940^3 = 1157590^3+1659770^3.
A272889
Cubefree taxi-cab numbers that are not squarefree.
Original entry on oeis.org
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
Offset: 1
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.
-
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, ", ")));
A273354
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.
Original entry on oeis.org
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.
A372296
Taxicab numbers that are sandwiched between nonsquarefree numbers.
Original entry on oeis.org
20683, 216125, 327763, 593047, 684019, 842751, 1845649, 2691451, 4505949, 4744376, 5004125, 5772403, 6058747, 7640128, 8029000, 8216000, 8494577, 10702783, 10765603, 10821896, 11859211, 12533824, 13731319, 14916727, 16776487, 18406603, 18617625, 20616463, 22031576, 24480125, 25937576, 27529073
Offset: 1
20683 = 13 * 37 * 43 (between 20682 = 2 * 3^3 * 383 and 20684 = 2^2 * 5171).
216125 = 5^3 * 7 * 13 * 19 (between 216124 = 2^2 * 71 * 761 and 216126 = 2 * 3^2 * 12007).
327763 = 31 * 97 * 109 (between 327762 = 2 * 3^2 * 131 * 139 and 327764 = 2^2 * 67 * 1223).
- Christian Boyer, Les nombres Taxicabs, in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
-
Select[Import["https://oeis.org/A001235/b001235.txt", "Table"][[;; , 2]], # < 3*10^7 && Nor @@ SquareFreeQ /@ (# + {-1, 1}) &] (* Amiram Eldar, Apr 25 2024 *)
A384106
Numbers representable as the sum of 2 cubes in at least 2 ways generated by a parameterized formula involving (7+4*sqrt(3))^n and (7-4*sqrt(3))^n.
Original entry on oeis.org
1009736, 2714690888, 7334904115448, 19818905563705976, 53550675461437475048, 144693905277386048024168, 390962878508814502873889816, 1056203940519850679825934312168, 2853755704387709706549646191448888, 7710144396612746633517746345789261976
Offset: 1
For n = 7, a(7) = x(n)^3 + y(n)^3 = ((-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^7 + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^7)/4 + 3)^3 + ((-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^7 + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^7)/4)^3 = 390962878508814502873889816.
Comments