A002760 -id:A002760 - cp's OEIS frontend

A075052 Write down squares and cubes in order, as in A002760. The sequence gives the first differences between terms.

1, 3, 4, 1, 7, 9, 2, 9, 13, 15, 17, 19, 21, 4, 19, 25, 27, 20, 9, 31, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 91, 81, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 111, 113, 115, 11

Offset: 1

Zak Seidov, Oct 07 2002

nonn

Ordered sequence of squares and cubes is as in A002760.

			First sorted squares and cubes are: 0, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, hence a(1) = 1 - 0 = 1, a(10) = 64 - 49 = 15, a(12) = 100 - 81 = 19.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A002760.

Module[{nn=70,sq,cb},sq=Range[0,nn]^2;cb=Select[Range[nn]^3,#<=sq[[-1]]&];Join[sq,cb]//Union//Differences] (* Harvey P. Dale, Jun 11 2024 *)

lista(nn) = {vec = vector(nn, i, i-1); pp = select(i->(issquare(i) || (ispower(i, 3))), vec); for (i=1, #pp-1, print1(pp[i+1] - pp[i], ", "););} \\ Michel Marcus, Oct 03 2013

from math import isqrt
from sympy import integer_nthroot
def A075052(n):
    def f(x): return n-1+x+integer_nthroot(x,6)[0]-(b:=integer_nthroot(x,3)[0])-(a:=isqrt(x)), a, b
    m = n-1
    k, a, b = f(n-1)
    while m != k:
        m = k
        k, a, b = f(k)
    return min((a+1)**2,(b+1)**3)-m # Chai Wah Wu, Aug 09 2024

a(n) = A002760(n+1) - A002760(n).

A374754 a(n) is the difference between the sum of the squares and the sum of the cubes for the n first terms of A002760.

Original entry on oeis.org

0, 0, 4, -4, 5, 21, 46, 19, 55, 104, 104, 185, 285, 406, 281, 425, 594, 790, 574, 799, 1055, 1344, 1668, 1325, 1686, 2086, 2527, 3011, 2499, 3028, 3604, 4229, 4905, 4905, 5689, 6530, 7430, 8391, 7391, 8415, 9504, 10660, 11885, 13181, 11850, 13219, 14663, 16184

Offset: 1

Felix Huber, Jul 28 2024

sign

For A002760(n) <= k < A002760(n+1), the difference between the sum of the squares and the sum of the cubes in the first k nonnegative integers is a(n).

			a(7) = a(6) + A002760(7) = 21 + 1*25 = 46, since 25 is a square but not a cube.
a(8) = a(7) - A002760(8) = 46 + (-1)*27 = 19, since 27 is a cube but not a square.
a(11) = a(10) + A002760(11) - A002760(11) = 104 + 0*64 = 104, since 64 is a square and a cube.
The difference between the sum of the squares and the sum of the cubes in the first 24 nonnegative integers is a(6) = 21, because A002760(6) = 16 <= 24 < A002760(7) = 25.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000330 (sum of squares), A000537 (sum of cubes), A001014 (sixth powers), A002760 (squares and cubes), A061023, A087285, A087286.

isA374754:=proc(k)
   option remember;
   if k=0 then 0
   elif issqr(k) and not type(root(k,3),integer) then procname(k-1)+k;
   elif type(root(k,3),integer) and not issqr(k) then procname(k-1)-k;
   else procname(k-1)
   fi;
end proc;
A374754:=k->
   if k=0 then 0
   elif isA374754(k)<>isA374754(k-1) or type(root(k,6),integer) then isA374754(k)
   fi;
seq(A374754(k),k=0..1521);

lista(nn) = my(v = select(x->issquare(x) || ispower(x, 3), [0..nn]), s=0, w = vector(#v)); for (i=1, #v, if (issquare(v[i]), s += v[i]); if (ispower(v[i], 3), s -= v[i]); w[i] = s;); w; \\ Michel Marcus, Aug 04 2024

from math import isqrt
from sympy import integer_nthroot
def A374754(n):
    def f(x): return n-1+x+integer_nthroot(x,6)[0]-(b:=integer_nthroot(x,3)[0])-(a:=isqrt(x)), a, b
    m = n-1
    k, a, b = f(n-1)
    while m != k:
        m = k
        k, a, b = f(k)
    return a*(a+1)*((a<<1)+1)//3-((b*(b+1))**2>>1)>>1 # Chai Wah Wu, Aug 09 2024

a(1) = 0. For n >= 2, a(n) = a(n-1) + f*A002760(n) where f = 1 if A002760(n) is a square but not a cube, f = -1 if A002760(n) is a cube but not a square and f = 0 if A002760(n) is a square and a cube.

A022549 Sum of a square and a nonnegative cube.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 12, 16, 17, 24, 25, 26, 27, 28, 31, 33, 36, 37, 43, 44, 49, 50, 52, 57, 63, 64, 65, 68, 72, 73, 76, 80, 81, 82, 89, 91, 100, 101, 108, 113, 121, 122, 125, 126, 127, 128, 129, 134, 141, 144, 145

Offset: 1

N. J. A. Sloane

It appears that there are no modular constraints on this sequence; i.e., every residue class of every integer has representatives here. - Franklin T. Adams-Watters, Dec 03 2009

A045634(a(n)) > 0. - Reinhard Zumkeller, Jul 17 2010

R. Zumkeller, Table of n, a(n) for n = 1..10000 - _Reinhard Zumkeller_, Jul 17 2010

Complement of A022550; A002760 and A179509 are subsequences.

Cf. A055394, A045634, A055393, A123291, A169618, A123364, A111925.

q=30; imax=q^2; Select[Union[Flatten[Table[x^2+y^3, {y,0,q^(2/3)}, {x,0,q}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)

is(n)=for(k=0,sqrtnint(n,3), if(issquare(n-k^3), return(1))); 0 \\ Charles R Greathouse IV, Aug 24 2020

list(lim)=my(v=List(),t); for(k=0,sqrtnint(lim\=1,3), t=k^3; for(n=0,sqrtint(lim-t), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Aug 24 2020

A077110 Nearest integer cube to n^2.

Original entry on oeis.org

0, 1, 1, 8, 8, 27, 27, 64, 64, 64, 125, 125, 125, 125, 216, 216, 216, 343, 343, 343, 343, 512, 512, 512, 512, 729, 729, 729, 729, 729, 1000, 1000, 1000, 1000, 1000, 1331, 1331, 1331, 1331, 1331, 1728, 1728, 1728, 1728, 1728, 2197, 2197, 2197

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(10) = 125, as 125 = 5^3 is the nearest cube to 100 = 10^2.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000578, A000290, A075847, A070923, A077106, A077107, A077111, A077118.

Cf. A002760 (Squares and cubes). - Zak Seidov, Oct 08 2015

nic[n_]:=Module[{n2=n^2,s3,c1,c2},s3=Surd[n2,3];c1=Floor[s3]^3;c2= Ceiling[ s3]^3;If[n2-c1Harvey P. Dale, Jul 05 2015 *)

from sympy import integer_nthroot
def A077110(n):
    n2 = n**2
    a = integer_nthroot(n2,3)[0]
    a2, a3 = a**3, (a+1)**3
    return a3 if a3+a2-2*n2 < 0 else a2 # Chai Wah Wu, Sep 24 2021

a(n) = if A075847(n) < A070923(n) then A077106(n) else A077107(n).

A131799 Number of partitions of n into parts that are squares or cubes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 4, 5, 5, 5, 7, 8, 8, 8, 12, 14, 15, 15, 19, 21, 22, 22, 28, 33, 35, 37, 43, 48, 50, 52, 61, 69, 74, 78, 90, 98, 103, 107, 122, 135, 143, 152, 170, 186, 194, 203, 225, 247, 261, 275, 305, 330, 348, 362, 396, 429, 454, 477, 519, 561, 590, 618, 666, 717

Offset: 0

Reinhard Zumkeller, Jul 16 2007

nonn

a(n) = A078635(n) for n < 32 = 2^5.

			a(10) = #{9+1, 8+1+1, 4+4+1+1, 4+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 5.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000041, A001156, A002760, A003108, A078635.

nmax = 65; c2max = nmax^(1/2); c3max = nmax^(1/3);
s = Flatten[{Table[n^2, {n, 1, c2max}]}~Join~{Table[n^3, {n, 1, c3max}]}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

G.f.: Product_{k>=1} (1 - x^(k^6)) / ((1 - x^(k^2)) * (1 - x^(k^3))). - Vaclav Kotesovec, Jan 12 2017

a(0)=1 prepended by Ilya Gutkovskiy, Jan 11 2017

A087797 Primes, squares of primes and cubes of primes.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Benoit Cloitre, Oct 10 2003

nonn

Union of A000040 and A168363. - Chai Wah Wu, Aug 09 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A000430, A002760, A168363.

m=400;Union[Prime[Range[PrimePi[m]]],Prime[Range[PrimePi[m^(1/2)]]]^2,Prime[Range[PrimePi[m^(1/3)]]]^3] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
With[{nn=70},Take[Union[Flatten[{#,#^2,#^3}&/@Prime[Range[nn]]]],nn]] (* Harvey P. Dale, Oct 16 2012 *)

is(n)=my(t=isprimepower(n)); t && t<4 \\ Charles R Greathouse IV, Oct 19 2015

from math import isqrt
from sympy import primepi, integer_nthroot
def A087797(n):
    def f(x): return n+x-primepi(x)-primepi(isqrt(x))-primepi(integer_nthroot(x,3)[0])
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

a(n) ~ n log n. - Charles R Greathouse IV, Oct 19 2015

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

Alex Ratushnyak, Dec 03 2013

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

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

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

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

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);
    }
  }
}

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 *)

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

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)

A377025 Squares and cubes that are not 6th powers.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 36, 49, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025

Offset: 1

Chai Wah Wu, Oct 13 2024

Squares and cubes that cannot be written as both a square and a cube.
{A002760}\{A001014}.
A125643 minus the repeated terms.

Cf. A002760, A125643, A001014.

lim=2025;Select[Union[Range[Floor[lim^(1/2)]]^2,Range[Floor[lim^(1/3)]]^3],!IntegerQ[#^(1/6)]&] (* James C. McMahon, Oct 16 2024 *)

from math import isqrt
from sympy import integer_nthroot
def A377025(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,3)[0]-isqrt(x)
    return bisection(f,n,n)

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

Original entry on oeis.org

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=',')

A125643 Squares and cubes (with repetition).

Original entry on oeis.org

0, 0, 1, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681

Offset: 1

Zak Seidov, Oct 19 2006

nonn

Repeating terms are sixth powers: 0,1,64,729,... (A001014).

For numbers not appearing as a difference between a square and an adjacent cube in this list, see A054504 and A081121.

Zak Seidov, Table of n, a(n) for n = 1..1000.

Cf. A002760 (squares and cubes (without repetitions)).

Cf. A001014, A054504, A081121, A087285, A087286, A088017.

m=1681;cm=Floor[m^(1/3)];sm=Floor[Sqrt[m]];s=Range[0,sm]^2;c=Range[0,cm]^3;Sort[Join[s,c]] (* James C. McMahon, Dec 20 2024 *)

from math import isqrt
from sympy import integer_nthroot
def A125643(n):
    if n <= 4: return n-1>>1
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-2+x-integer_nthroot(x,3)[0]-isqrt(x)
    return bisection(f,n-2,n-2) # Chai Wah Wu, Oct 14 2024

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007

cp's OEIS Frontend

A075052 Write down squares and cubes in order, as in A002760. The sequence gives the first differences between terms.

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A374754 a(n) is the difference between the sum of the squares and the sum of the cubes for the n first terms of A002760.

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A022549 Sum of a square and a nonnegative cube.

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A077110 Nearest integer cube to n^2.

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A131799 Number of partitions of n into parts that are squares or cubes.

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A087797 Primes, squares of primes and cubes of primes.

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

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