cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Alex Ratushnyak, Dec 03 2013

Keywords

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

Examples

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

Links

Crossrefs

Cf. A000290, A000578, A075454, A233074.
Cf. A002760 (Squares and cubes).
Cf. A001014 (Additional terms if k2=k3 were allowed).

Programs

  • 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);
    }
  }
}
  • 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 *)
  • 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(m2n3, n3=n++^3, m2=m++^2; n3=n++^3); s=m2+n3); Vec(v) \\ Charles R Greathouse IV, Jul 29 2016
  • 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)

A249875 Numbers that are exactly halfway between the nearest square and the nearest power of 2.

Original entry on oeis.org

3, 6, 34, 136, 498, 2082, 8146, 32946, 131058, 524232, 2096928, 8387712, 33550848, 134226562, 536859906, 2147439624, 8589943858, 34359775432, 137439101728, 549756406912, 2199022661826, 8796090647304, 35184374452498, 140737497809992, 562949943786834, 2251799775147336
Offset: 1

Views

Author

Alex Ratushnyak, Nov 07 2014

Keywords

Comments

Numbers that are the arithmetic mean of the nearest square and the nearest power of 2 (other than that nearest square).

Examples

			3 is a term because 2<3<4; 6 is a term because 4<6<8.

Links

Crossrefs

Cf. A000290, A000079, A233074, A233075.

Programs

  • Python
    def isqrt(a):
    sr = 1 << (a.bit_length() >> 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 j in range(99):
    i = 2**j
    r = isqrt(i)
    if r * r == i:
        continue
    if r & 1:
        a = ((r + 1) * (r + 1) + i) // 2
    else:
        a = (i + r * r) // 2
    print(a, end=', ')
  • Python
    from gmpy2 import isqrt
A249875_list, x = [], 1
for _ in range(10**3):
    A249875_list.append(2*sum(divmod(isqrt(2*x),2))**2+x)
    x *= 4 # Chai Wah Wu, Dec 16 2014

A233443 Primes that are exactly between the nearest square and the nearest triangular number.

Original entry on oeis.org

2, 5, 23, 47, 193, 389, 1667, 8807, 9431, 10177, 10597, 10847, 11831, 13411, 17183, 22433, 29201, 33893, 36073, 38321, 40093, 42461, 48991, 50131, 54287, 54851, 57037, 63347, 65183, 67121, 71917, 87803, 88607, 91291, 94847, 104491, 108293, 112163, 116101, 117167, 122033
Offset: 1

Views

Author

Alex Ratushnyak, Dec 09 2013

Keywords

Comments

A subsequence of A233074.

Crossrefs

Cf. A000040, A000217, A000290, A233074.

Programs

  • Mathematica
    nearestTri[n_] := Block[{a = Floor@ Sqrt[ 2n - 1]}, If[ 4n < a (a + 3), a (a - 1)/2, a (a + 1)/2]]; nearestSq[n_] :=  Block[{a = Floor@ Sqrt@ n}, If[a^2 + a + 1/2 > n, a^2, a^2 + 2 a + 1]]; Select[ Prime@ Range@ 12000, 2# == nearestSq@# + nearestTri@# &] (* Robert G. Wilson v, Aug 01 2014 *)
  • PARI
    lista(nn) = {forprime(p=2, nn, sqp = sqrtint(p); ps = sqp^2; ns = (sqp+1)^2; sqt = floor((sqrt(8*p+1) - 1)/2); pt = sqt*(sqt+1)/2; nt = (sqt+2)*(sqt+1)/2; if (((ds=p-ps) < (ns-p)) && ((dt=(nt-p)) <= p-pt) && (ds == dt), print1(p, ", "), if (((ds=ns-p) < (p-ps)) && ((dt=(p-pt)) < nt-p) && (ds == dt), print1(p, ", "));););} \\ Michel Marcus, Aug 11 2014

Extensions

Corrected by Alex Ratushnyak, Jun 08 2014
Showing 1-3 of 3 results.

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