A233075 Numbers that are midway between the nearest square and the nearest cube.

6, 26, 123, 206, 352, 498, 1012, 1350, 1746, 2203, 2724, 3428, 4977, 5804, 6874, 8050, 9335, 10732, 12244, 13874, 17500, 19782, 21928, 24519, 26948, 29860, 32946, 35829, 39254, 42862, 50639, 54814, 59184, 63752, 69045, 74036, 79234, 85224, 90863, 97340, 104076

Offset

1,1

Comments

The sequence of roots of nearest squares begins: 2, 5, 11, 14, 19, 22, 32, 37, 42, 47, 52, 59, 71, 76, 83, 90, 97, 104, 111, 118, 132, ...

The sequence of cube roots of nearest cubes begins: 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, ... (Cf. A000037)

The sequence of k-k2 (equals k3-k) begins: 2, 1, 2, 10, -9, 14, -12, -19, -18, -6, 20, -53, -64, 28, -15, -50, -74, -84, -77, -50, ...

If we allow k2=k3 then first missing terms are 0, 1, 64, 729, 4096, ... . - Zak Seidov, Dec 10 2013

Links

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2108 from Seidov)

Example

26 = 5^2 + 1 = 3^3 - 1.
352 = 19^2 - 9 = 7^3 + 9.

Mathematica

max = 10^6; u = Union[Range[Ceiling[Sqrt[max]]]^2, Range[Ceiling[ max^(1/3) ]]^3]; Reap[Do[x = u[[k]]; y = u[[k+1]]; If[Not[IntegerQ[Sqrt[x]] && IntegerQ[Sqrt[y]]] && Not[IntegerQ[x^(1/3)] && IntegerQ[y^(1/3)]] && IntegerQ[m = (x+y)/2], Sow[m]], {k, 1, Length[u]-2}]][[2, 1]] (* Jean-François Alcover, Dec 03 2015 *)
Module[{upto=150000, nns}, nns=Union[Join[Range[Floor[Sqrt[upto]]]^2, Range[Floor[Surd[upto, 3]]]^3]]; Mean/@Select[Partition[nns, 2, 1], EvenQ[Total[#]]&]] (* Harvey P. Dale, Nov 06 2017 *)

Prog

(Java)
import java.math.*;
public class A233075 {
  public static void main (String[] args) {
    for (long k = 1; ; k++) { // ok for small k's
      long r2=(long)Math.sqrt(k), r3=(long)Math.cbrt(k);
      long b2=r2*r2, a2=b2+r2*2+1; //squares below and above
      long b3=r3*r3*r3, a3=b3+3*r3*(r3+1)+1; //cubes below, above
      if ((b2+a3==k*2 && k-b2<=a2-k && a3-k<=k-b3) ||
          (b3+a2==k*2 && k-b3<=a3-k && a2-k<=k-b2))
            System.out.printf("%d, ", k);
    }
  }
}
(Python)
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
a=[]
for c in range(1, 10000):
    cube = c*c*c
    srB = isqrt(cube)
    srB2= srB**2
    if srB2==cube: continue
    if ((srB2^cube)&1)==0:
        n = (srB2+cube)//2
    else:
        n = (srB2+2*srB+1+cube)//2
    a.append(n)
print(a)
(PARI) list(lim)=my(v=List(), m=2, n=2, m2=4, n3=8, s=12); lim*=2; while(s <= lim, if(s%2==0 && m2!=n3 && abs(s/2-m2)<=abs(s/2-(m-1)^2) && abs(s/2-m2)<=abs(s/2-(m+1)^2) && abs(s/2-m2)<=abs(s/2-(n-1)^3) && abs(s/2-m2)<=abs(s/2-(n+1)^3), listput(v, s/2)); if(m2<n3, m2=m++^2, m2>n3, n3=n++^3, m2=m++^2; n3=n++^3); s=m2+n3); Vec(v) \\ Charles R Greathouse IV, Jul 29 2016

Crossrefs

Cf. A000290, A000578, A075454, A233074.

Cf. A002760 (Squares and cubes).

Cf. A001014 (Additional terms if k2=k3 were allowed).

Sequence in context: A164549 A283341 A046647 * A307331 A298625 A046233

Adjacent sequences: A233072 A233073 A233074 * A233076 A233077 A233078

Keyword

nonn,nice,easy

Author

Alex Ratushnyak, Dec 03 2013

Status

approved