A253181 -id:A253181 - cp's OEIS frontend

A268509 Numbers x such that x^3 = y^2 + z for some y and some nonzero z with -x < z < x.

2, 3, 5, 13, 15, 17, 32, 35, 37, 40, 43, 46, 52, 56, 63, 65, 99, 101, 109, 136, 143, 145, 152, 158, 175, 190, 195, 197, 243, 255, 257, 312, 317, 323, 325, 331, 336, 351, 356, 366, 377, 399, 401, 422, 483, 485, 560, 568, 575, 577, 584, 592, 654, 675, 677, 717, 741, 783, 785, 799, 810, 891, 899, 901, 909, 937, 944, 978

Offset: 1

Daniel Mondot, Feb 06 2016

nonn

List of x such as x^3 is a near square (see examples).

Note that z = 17 appears often (see A029728).

			2^3 = 3^2 - 1;
3^3 = 5^2 + 2;
5^3 = 11^2 + 4;
13^3 = 47^2 - 12;
15^3 = 58^2 + 11;
17^3 = 70^2 + 13;
32^3 = 181^2 + 7;
35^3 = 207^2 + 26;
37^3 = 225^2 + 28;
40^3 = 253^2 - 9;
43^3 = 282^2 - 17;
46^3 = 312^2 - 8;
52^3 = 375^2 - 17;
56^3 = 419^2 + 55;
63^3 = 500^2 + 47;
65^3 = 524^2 + 49;
99^3 = 985^2 + 74.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A029728, A253181, A268510.

#include 
#include 
#include 
#define MAX2 10000
/* list number x and y such that x^3 = y^2 ± delta (0 < delta < x) */
/* this generates A268509 and A268510 */
long long unsigned b,c,d;
long long signed ds;
unsigned long long list2[MAX2];
unsigned long long list3[MAX2];
long double b1, cd, dd;
void main(unsigned argc, char *argv[])
{
unsigned a, i;
  i=0;
  // I never actually calculate b^3 or c^2, but only b^(3/2) = c + ds
  // this allows me to indirectly check b^3 past 2^64
  for (b=0; b<100000000; ++b) // could go up to b<4294967295u; max
  {
    b1 = sqrtl(b);
    cd= b1 *(long double)b;
    c=(long long unsigned)(cd+(double)0.5);
    dd = 2 * c * (cd - c);
    if (dd<0) ds = (dd - 0.5);
    else ds = (dd + 0.5);
    d = llabs(ds);
    if (dA268509 */
  for (a=0; aA268510 */
  for (a=0; a

is(n)=my(t=abs(n^3-round(n^1.5)^2)); 0Charles R Greathouse IV, Feb 09 2016

A268510 Numbers x such that x^2 = y^3 + z (0 < abs(z) < y).

Original entry on oeis.org

3, 5, 11, 47, 58, 70, 181, 207, 225, 253, 282, 312, 375, 419, 500, 524, 985, 1015, 1138, 1586, 1710, 1746, 1874, 1986, 2315, 2619, 2723, 2765, 3788, 4072, 4120, 5511, 5644, 5805, 5859, 6022, 6159, 6576, 6717, 7002, 7320, 7970, 8030, 8669, 10615, 10681, 13252, 13537, 13788, 13860, 14113, 14404, 16725, 17537, 17615

Offset: 1

Daniel Mondot, Feb 06 2016

nonn

List of n such as n^2 is a near cube (see examples).
Numbers x such that x^2 = y^3 + 0 (e.g. 1000^2 = 100^3) are omitted.
Note that a delta of 17 appears often. See A029728.

			3^2 = 2^3 + 1
5^2 = 3^3 - 2
11^2 = 5^3 - 4
47^2 = 13^3 + 12
58^2 = 15^3 - 11
70^2 = 17^3 - 13
181^2 = 32^3 - 7
207^2 = 35^3 - 26
225^2 = 37^3 - 28
253^2 = 40^3 + 9
282^2 = 43^3 + 17
312^2 = 46^3 + 8
375^2 = 52^3 + 17
419^2 = 56^3 - 55
500^2 = 63^3 - 47
524^2 = 65^3 - 49
985^2 = 99^3 - 74

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A029728, A268509, A253181.

Select[Range@ 5000, Resolve@ Exists[{y, z}, And[Reduce[#^2 == (y^3 + z), {y, z}, Integers], 0 < Abs@ z < y]] &] (* Michael De Vlieger, Feb 07 2016 *)

cp's OEIS Frontend

A268509 Numbers x such that x^3 = y^2 + z for some y and some nonzero z with -x < z < x.

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