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A233075 Numbers that are midway between the nearest square and the nearest cube. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 27 16:48 EDT 2022. Contains 354110 sequences. (Running on oeis4.)