A077119 -id:A077119 - cp's OEIS frontend

A077116 n^3 - A065733(n).

0, 0, 4, 2, 0, 4, 20, 19, 28, 0, 39, 35, 47, 81, 40, 11, 0, 13, 56, 135, 79, 45, 39, 67, 135, 0, 152, 83, 48, 53, 104, 207, 7, 216, 100, 26, 0, 28, 116, 270, 496, 277, 104, 546, 503, 524, 615, 139, 368, 0, 391, 155, 732, 652, 648, 726, 55, 293, 631, 170, 704

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) = 0 for n = m^2. - Zak Seidov, May 11 2007

It has been asked whether some primes do not occur in this sequence. It seems indeed that primes 3, 5, 17, 23, 29, 31, 37, 41, 43, 59, 61,... do not occur, primes 2, 7, 11, 13, 19, 47, 53, 67, 79, 83,... do. For further investigations, see A087285 = the range of this sequence, and also the related sequences A229618 = range of A181138, and A165288. - M. F. Hasler, Sep 26 2013 and Oct 05 2013

			A065733(10) = 961 = 31^2 is the largest square less than or equal to 10^3 = 1000, therefore a(10) = 1000 - 961 = 39.

Cf. A000578, A070929, A077118, A077119, A075847.

A077116 := proc(n)
    A053186(n^3) ;
end proc: # R. J. Mathar, Jul 12 2016

Table[c = n^3; c - Floor[Sqrt[c]]^2, {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 02 2011 *)

A077116(n)=n^3-sqrtint(n^3)^2 \\ - M. F. Hasler, Sep 26 2013

a(n) = A154333(n) unless n is a square or, equivalently, a(n)=0. - M. F. Hasler, Oct 05 2013

a(n) = A053186(n^3). - R. J. Mathar, Jul 12 2016

A077118 Nearest integer square to n^3.

Original entry on oeis.org

0, 1, 9, 25, 64, 121, 225, 361, 529, 729, 1024, 1296, 1764, 2209, 2704, 3364, 4096, 4900, 5776, 6889, 7921, 9216, 10609, 12100, 13924, 15625, 17689, 19600, 21904, 24336, 26896, 29929, 32761, 36100, 39204, 42849, 46656, 50625, 54756, 59536

Offset: 0

Reinhard Zumkeller, Oct 29 2002

nonn

			a(5)=121, as 121=11^2 is the nearest square to 125=5^3.

Cf. A000290, A000578, A002821, A077119, A077110.

Table[Round[Sqrt[n^3]]^2, {n, 0, 39}] (* Alonso del Arte, Dec 07 2011, based on Artur Jasinski's program for A077119 *)

from math import isqrt
def A077118(n): return ((m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1))**2 # Chai Wah Wu, Jul 29 2022

a(n) = if A077116(n) < A070929(n) then A065733(n) else A077115(n).

a(n) = A002821(n)^2. - Chai Wah Wu, Jul 30 2022

A200656 Successive values x such that the Mordell elliptic curve x^3 - y^2 = d has extremal points with quadratic extension over the rationals.

Original entry on oeis.org

1942, 2878, 3862, 6100, 8380, 11512, 15448, 18694, 31228, 93844, 111382, 117118, 129910, 143950, 186145, 210025, 375376, 445528, 468472, 575800, 844596, 1002438, 1054062, 1193740, 1248412, 1326025, 1388545, 1501504, 1697908, 1782112, 1813660, 1873888, 1946737

Offset: 1

Artur Jasinski, Nov 20 2011

nonn

Definition: Extremal points on the Mordell elliptic curve x^3 - y^2 = d are points (x,y) such that x^3 - round(sqrt(x^3))^2 = d. For values d for successive x independent of the extensions see A077119.
For y values see A200657.
For d values see A200658.
Definition: Secondary terms occur when there exist integers k such that A200656 is divisible by k^2, A200657 is divisible by k^3 and A200658 is divisible by k^6.
Terms free of such k are primary terms; see A201047. Secondary terms are, e.g., a(6)=a(2)*2^2, a(7)=a(3)*2^2, a(17)=a(10)*2^2, a(18)=a(11)*2^2, a(19)=a(12)*2^2, a(21)=a(10)*3^2.
For successive secondary terms, see A201048.
A200216 is a subsequence of this sequence.

Peter J. C. Moses and Artur Jasinski, Table of n, a(n) for n = 1..185 (all terms below 10^9)

Cf. A200216, A201047, A201048.

Cf. A077119, A200657, A200658.

A070929 Smallest integer >= 0 of the form x^2 - n^3.

Original entry on oeis.org

0, 0, 1, 9, 0, 19, 9, 18, 17, 0, 24, 38, 36, 12, 65, 106, 0, 128, 97, 30, 100, 148, 168, 154, 100, 0, 113, 198, 249, 260, 225, 138, 356, 163, 297, 389, 0, 423, 353, 217, 9, 248, 441, 17, 80, 79, 8, 506, 297, 0, 316, 574, 17, 119, 145, 89, 784, 568, 252, 737, 225, 548

Offset: 0

Benoit Cloitre, May 20 2002

a(n)=0 iff n is a square.

			A077115(10) = 1024 = 32^2 is the least square >= 10^3 = 1000, therefore a(10) = 1024 - 1000 = 24.

Cf. A000578, A077116, A077118, A077119, A070923, A068527.

f[n_]=Ceiling[n^(3/2)]^2-n^3;
t1=Table[f[n], {n, 1, 90}]; t1 (* Clark Kimberling, Jan 30 2011 *)

for(n=1,100,print1(ceil(n^(3/2))^2-n^3,","))

a(n) = ceiling(n^(3/2))^2 - n^3 = A077115(n) - n^3.

A077111 a(n) = A077110(n) - n^2.

Original entry on oeis.org

0, 0, -3, -1, -8, 2, -9, 15, 0, -17, 25, 4, -19, -44, 20, -9, -40, 54, 19, -18, -57, 71, 28, -17, -64, 104, 53, 0, -55, -112, 100, 39, -24, -89, -156, 106, 35, -38, -113, -190, 128, 47, -36, -121, -208, 172, 81, -12, -107, -204, 244, 143, 40, -65, -172

Offset: 0

Reinhard Zumkeller, Oct 29 2002

sign

a(n)=0 iff n = m^(6*k).

			A077110(20)=343=7^3 is the nearest cube to 20^2=400, therefore a(20)=343-400=-57.

Cf. A000290, A077110, A077119.

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

A002938 The minimal sequence from solving n^3 - m^2 = a(n).

Original entry on oeis.org

0, 1, 2, 0, 4, 9, 18, 17, 0, 24, 35, 36, 12, 40, 11, 0, 13, 56, 30, 79, 45, 39, 67, 100, 0, 113, 83, 48, 53, 104, 138, 7, 163, 100, 26, 0, 28, 116, 217, 9, 248, 104, 17, 80, 79, 8, 139, 297, 0, 316, 155, 17, 119, 145, 89, 55, 293, 252, 170, 225, 405, 184, 47, 0, 49

Offset: 1

N. J. A. Sloane

nonn

Marshall Hall, Jr., The Diophantine equation x^3-y^2=k, pp. 173-198 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A077119.

f1[n_] := n - Floor[Sqrt[n]]^2;
f2[n_] := Ceiling[Sqrt[n]]^2-n;
Table[Min[f1[n^3], f2[n^3]], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2010 *)

a(n)=vecmin(vector(ceil(n^(3/2)),i,abs(n^3-i^2)))

a(n) = |A077119(n+1)|.

a(n^2) = 0. - Benoit Cloitre, Aug 17 2002

More terms from Benoit Cloitre, Aug 17 2002

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=',')

A233149 a(n) = ((n^2+1)^3) - s, where s is the nearest square to (n^2+1)^3.

Original entry on oeis.org

-1, 4, -24, 13, -113, 28, -316, 49, -681, 76, -1256, 109, -2089, 148, -3228, 193, -4721, 244, -6616, 301, -8961, 364, -11804, 433, -15193, 508, -19176, 589, -23801, 676, -29116, 769, -35169, 868, -42008, 973, -49681, 1084, -58236, 1201, -67721, 1324, -78184, 1453, -89673, 1588, -102236, 1729, -115921, 1876

Offset: 1

Artur Jasinski, Dec 05 2013

			Table of n, n^2, n^2+1, (n^2+1)^3, closest square, difference:
  1 1 2 8 9 -1
  2 4 5 125 121 4
  3 9 10 1000 1024 -24
  4 16 17 4913 4900 1
  ...

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A077119, A056107.

aa = {}; Do[AppendTo[aa, (n^2 + 1)^3 - Round[Sqrt[(n^2 + 1)^3]]^2], {n, 1, 50}]; aa
LinearRecurrence[{0,4,0,-6,0,4,0,-1},{-1,4,-24,13,-113,28,-316,49},50] (* Harvey P. Dale, Aug 09 2025 *)

a(n) = (n^2+1)^3 - (round(sqrt((n^2+1)^3)))^2.
Recurrence formula: a(n)= - a(n-2) + 4*a(n-4) - 6*a(n-6) + 4*a(n-8).
a(n) = -A077119(n^2+1). - R. J. Mathar, Jan 18 2021
a(2*n) = A056107(n). - R. J. Mathar, Jan 18 2021

cp's OEIS Frontend

A077116 n^3 - A065733(n).

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A077118 Nearest integer square to n^3.

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A200656 Successive values x such that the Mordell elliptic curve x^3 - y^2 = d has extremal points with quadratic extension over the rationals.

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A070929 Smallest integer >= 0 of the form x^2 - n^3.

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A077111 a(n) = A077110(n) - n^2.

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A002938 The minimal sequence from solving n^3 - m^2 = a(n).

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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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A233149 a(n) = ((n^2+1)^3) - s, where s is the nearest square to (n^2+1)^3.

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