A268510 -id:A268510 - cp's OEIS frontend

A253181 Numbers n such that the distance between n^3 and the nearest square is less than n.

1, 2, 3, 4, 5, 9, 13, 15, 16, 17, 25, 32, 35, 36, 37, 40, 43, 46, 49, 52, 56, 63, 64, 65, 81, 99, 100, 101, 109, 121, 136, 143, 144, 145, 152, 158, 169, 175, 190, 195, 196, 197, 225, 243, 255, 256, 257, 289, 312, 317, 323, 324, 325, 331, 336, 351, 356, 361, 366, 377

Offset: 1

Alex Ratushnyak, Mar 23 2015

nonn

Distance can be zero, that is, cubes that are squares are included.

Numbers n such that A002938(n) < n.

			The distance between 5^3=125 and the nearest square 11^2=121 is less than 5, so 5 is in the sequence.

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

Cf. A000290, A000578, A002760.
Cf. A116885, A002938, A077119.
Cf. A268509, A268510.

dnsQ[n_]:=Module[{n3=n^3,sr},sr=Sqrt[n3];Min[n3-Floor[sr]^2, Ceiling[ sr]^2- n3]Harvey P. Dale, Dec 23 2015 *)

def isqrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
        s = sr + b
        if a >= s*s:  sr = s
        b>>=1
    return sr
for n in range(1000):
    cube = n*n*n
    r = isqrt(cube)
    sqr = r**2
    if cube-sqr < n or sqr+2*r+1-cube < n:  print(str(n), end=',')

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

Original entry on oeis.org

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

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A253181 Numbers n such that the distance between n^3 and the nearest square is less than n.

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