A077116 -id:A077116 - cp's OEIS frontend

A154101 Numbers n with property that A077116(n) is nonzero square.

2, 5, 13, 34, 65, 130, 185, 193, 205, 260, 338, 482, 865, 901, 970, 1325, 1352, 1450, 1469, 1490, 1825, 1928, 2701, 3077, 3604, 3757, 4234, 4250, 4637, 5245, 5465, 5525, 6649, 6722, 7300, 7514, 7585, 7930, 8066, 8109, 10370, 10685, 10804, 11125, 12545

Offset: 1

Zak Seidov, Jan 04 2009

nonn

A077116(a(n)) = A154101(n)^2.

			a(1) = 2 because A077116(2) = A154101(1)^2 = 4, a(3) = 13 because A077116(13) = A154101(3)^2 = 81.

Cf. A154100, A077116.

A154100 Numbers n with property that n^2 are nonzero square terms in A077116.

Original entry on oeis.org

2, 2, 9, 10, 7, 26, 37, 36, 30, 56, 26, 38, 145, 26, 170, 265, 208, 198, 130, 298, 73, 304, 135, 362, 208, 663, 730, 238, 767, 510, 704, 350, 915, 142, 584, 850, 369, 1090, 1090, 702, 730, 95, 1080, 1245, 97, 802, 1664, 738, 1090, 1971, 1904, 1276, 1295, 738, 1394

Offset: 1

Zak Seidov, Jan 04 2009

nonn

			2^2=4=A077116(2), 2^2=4=A077116(5), 9^2=81=A077116(13), 10^2=100=A077116(34).

Cf. A154101 corresponding indices of terms in A077116: 2,5,13,34,...

A179388 Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

Original entry on oeis.org

5, 11, 181, 207, 225, 500, 524, 1586, 13537, 376601, 223063347, 911054064, 16073515093, 22143115844, 29448160810, 1661699554612, 2498973838515, 26588790747913, 27582731314539, 178638660622364

Offset: 1

Artur Jasinski, Jul 12 2010, Jul 13 2010, Aug 03 2010

"Records of minima" means values A179386(n)=A154333(x) such that A154333(x') > A154333(x) for all x' > x, or equivalently A181138(y) such that A181138(y') > A181138(y) for all y' > y. See the main entry A179386 for all further considerations. - M. F. Hasler, Sep 30 2013
For d values see A179386, for x values see A179387.
Theorem (Artur Jasinski):
For any positive number x >= A179387(n), the distance between the cube of x and the square of any y (with x<>n^2 and y<>n^3) can't be less than A179386(n).
Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable there can't exist any such x (or the related y).

Cf. A179107, A179108, A179109, A179387, A179388.

Cf. A181138, A229618, A077116, A106265, A165288.

max = 1000; vecd = Table[10100, {n, 1, max}]; vecx = Table[10100, {n, 1, max}]; vecy = Table[10100, {n, 1, max}]; len = 1; min = 10100; Do[m = Floor[(n^3)^(1/2)]; k = n^3 - m^2; If[k != 0, If[k <= min, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; min = 10100; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m]], {n, 1, 13333677}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; yy (*Artur Jasinski*)

A179388(n) = sqrt(A179387(n)^3 - A179386(n)).

Edited by M. F. Hasler, Sep 30 2013

A065733 Largest square <= n^3.

Original entry on oeis.org

0, 1, 4, 25, 64, 121, 196, 324, 484, 729, 961, 1296, 1681, 2116, 2704, 3364, 4096, 4900, 5776, 6724, 7921, 9216, 10609, 12100, 13689, 15625, 17424, 19600, 21904, 24336, 26896, 29584, 32761, 35721, 39204, 42849, 46656, 50625, 54756, 59049, 63504

Offset: 0

Labos Elemer, Nov 15 2001

			a(10) = 961, as 961 = 31^2 is the largest square <= 1000 = 10^3.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000093, A048760, A065730-A065741, A087285.

Cf. A077115, A077116, A077118, A077121.

a065733 n = head [x | x <- reverse [0.. n^3], a010052 x == 1] -- Reinhard Zumkeller, Oct 10 2013

Table[Floor[Sqrt[w^3]//N]^2, {w, 1, 50}]

A065733(n)=sqrtint(n^3)^2  \\ M. F. Hasler, Oct 05 2013

a(n) + A077116(n) = n^3.
a(n) = A048760(n^3).
n^3 - 2*n^(3/2) <= a(n) <= n^3. - Charles R Greathouse IV, Dec 05 2022
a(n) = A000093(n)^2. - Amiram Eldar, Jul 14 2024

A075847 Difference between n^2 and the largest cube <= n^2.

Original entry on oeis.org

0, 0, 3, 1, 8, 17, 9, 22, 0, 17, 36, 57, 19, 44, 71, 9, 40, 73, 108, 18, 57, 98, 141, 17, 64, 113, 164, 0, 55, 112, 171, 232, 24, 89, 156, 225, 296, 38, 113, 190, 269, 350, 36, 121, 208, 297, 388, 12, 107, 204, 303, 404, 507, 65, 172, 281, 392, 505, 620, 106, 225, 346

Offset: 0

Zak Seidov and Reinhard Zumkeller, Oct 15 2002

a(n) = n^2 - A077106(n).
a(n) = 0 iff n = m^(6*k).
a(n) = 0 when n is a cube. See A070923.

			a(4)=8 because 4^2 - 2^3 = 8; a(9)=17 because 9^2 - 4^3 = 17.
A077106(20) = 343 = 7^3 is the largest cube <= 20^2 = 400, therefore a(20) = 400 - 343 = 57.

Zak Seidov, Table of n, a(n) for n = 0..2000

Cf. A000290, A070923, A077110, A077111, A077116.

Cf. A070923.

Table[a = n^2; a - Floor[a^(1/3)]^3, {n, 0, 300}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

Edited by N. J. A. Sloane at the suggestion of Zak Seidov, Oct 30 2008

A087285 Possible differences between a cube and the next smaller square.

Original entry on oeis.org

2, 4, 7, 11, 13, 15, 19, 20, 26, 28, 35, 39, 40, 45, 47, 48, 49, 53, 55, 56, 60, 63, 67, 74, 76, 79, 81, 83, 100, 104, 107, 109, 116, 127, 135, 139, 146, 147, 148, 150, 152, 155, 170, 174, 180, 184, 186, 191, 193, 200, 207, 212, 215, 216, 233, 235, 242, 244, 249

Offset: 1

Hugo Pfoertner, Sep 18 2003

nonn

Sequence and program were provided by Ralf Stephan Aug 28 2003.
Comment from David W. Wilson, Jan 05 2009: I believe there is an algorithm for solving x^3 - y^2 = k, which should have a finite number of solutions for any k. That means that we should in principle be able to compute this sequence.
Up to the initial 0 in A165288, these two sequences appear to be the same, but according to its current definition, A165288 should be the same as the (different) sequence A229618 = the range of the sequence A181138 (= least k>0 such that n^2+k is a cube): If n^2+k=y^3 is the smallest cube above n^2, then n^2 is not necessarily the largest square below y^3. E.g., 18 is in A181138 and A229618, since 9+18=27 is the least cube above 9=3^2, but 25=5^2 is the largest square below 27. - M. F. Hasler, Oct 05 2013

			a(1)=2 because the next smaller square below 3^3=27 is 5^2=25.

See under A081121.

Cf. A087286, A088017, A081121, A077116, A065733.

v=vector(200):for(n=2,10^7,t=n^3:s=sqrtint(t)^2: if(s==t,s=sqrtint(t-1)^2):tt=t-s: if(tt>0&&tt<=200&&!v[tt],v[tt]=n)):for(k=1,200,if(v[k],print1(k",")))

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

A077119 a(n) = A077118(n) - n^3.

Original entry on oeis.org

0, 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

Offset: 0

Reinhard Zumkeller, Oct 29 2002

sign

a(n)=0 iff n = m^(6*k).
Values d=x^3-y^2 of extremal points of elliptic Mordell curves. Definition for extremal points see A200656. Each value x has only one value of distance d when coordinate x is extremal point, but for many fixed distances d, the elliptic curve has more than 1 extremal point. - Artur Jasinski, Nov 30 2011
Theorem (Artur Jasinski): If a(n)>0 then a(n)<(4n^(3/2)-1)/4 for every n. If a(n)<0 then a(n)>(-4n^(3/2)-1)/4 for every n. a(n)=0 then n is perfect square. - Artur Jasinski, Dec 08 2011

			A077118(10)=1024=32^2 is the nearest square to 10^3=1000, therefore a(10)=1024-1000=24.

Cf. A000578, A077118, A077111.

|a(n)| = A002938(n).

[Round(Sqrt(n^3))^2-n^3: n in [0..60]]; // Vincenzo Librandi, Mar 24 2015

A077119 := proc(n)
    (round( sqrt(n^3) ))^2-n^3 ;
end proc: # R. J. Mathar, Jan 18 2021

Table[Round[Sqrt[x^3]]^2 - x^3, {x, 0, 100}]  (* Artur Jasinski, Nov 30 2011 *)

from math import isqrt
def A077119(n): return ((m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1))**2-k # Chai Wah Wu, Jul 29 2022

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

A181138 Least positive integer k such that n^2 + k is a cube.

Original entry on oeis.org

1, 7, 4, 18, 11, 2, 28, 15, 61, 44, 25, 4, 72, 47, 20, 118, 87, 54, 19, 151, 112, 71, 28, 200, 153, 104, 53, 271, 216, 159, 100, 39, 307, 242, 175, 106, 35, 359, 284, 207, 128, 47, 433, 348, 261, 172, 81, 535, 440, 343, 244, 143, 40, 566, 459, 350, 239, 126, 11

Offset: 0

Jason Earls, Oct 06 2010

a(n) = A070923(n) if n is not cube. Zak Seidov, Mar 26 2013

See A229618 for the range of this sequence. A179386 gives the range of b(n) = min{ a(m); m >= n }. The indices of jumps in this sequence are given in A179388 = { n | a(m)>a(n) for all m > n } = { 0, 5, 11, 181, 207, 225, 500, 524, 1586, ... }. - M. F. Hasler, Sep 26 2013

			a(11) = 4 because 11^2 + k is never a cube for k < 4, but 11^2 + 4 = 5^3. - _Bruno Berselli_, Jan 29 2013

Bruno Berselli, Table of n, a(n) for n = 0..1000 (Corrected Jan 19 2019)

Cf. A070923, A077116.

S:=[];
k:=1;
for n in [0..60] do
   while not IsPower(n^2+k,3) do
        k:=k+1;
   end while;
   Append(~S, k);
   k:=1;
end for;
S;  // Bruno Berselli, Jan 29 2013

Table[(1 + Floor[n^(2/3)])^3 - n^2, {n, 100}] (* Zak Seidov, Mar 26 2013 *)

A181138(n)=(sqrtnint(n^2,3)+1)^3-n^2 \\ Charles R Greathouse IV, Mar 26 2013

a(n) << n^(4/3). - Charles R Greathouse IV, Mar 26 2013

Extended to a(0)=1 by M. F. Hasler, Sep 26 2013

A165288 Possible values of the difference between a cube and the largest square not larger than the cube.

Original entry on oeis.org

0, 2, 4, 7, 11, 13, 19, 20, 26, 28, 35, 39, 40, 45, 47, 48, 49, 53, 55, 56, 60, 63, 67, 74, 76, 79, 81, 83, 100, 104, 107, 109, 116, 135, 139, 146, 147, 148, 150, 152, 155, 170, 174, 180, 184, 186, 191, 193, 200, 207, 212, 215, 216, 233, 235, 242, 244, 251, 270, 277

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

nonn

The values of A077116, sorted and duplicates removed.
Note that the values have been generated with a finite search radius and are not proved to be complete. [R. J. Mathar, Oct 09 2009]
Except for the leading 0, a subsequence of A229618 which is in turn (except for the initial 1) a subsequence of A106265.  The values {15, 18, 25, 44, 54, 61, 71, 72, 87, 106, 112, 118, 126, 127,...} are in  A229618 but not in the present sequence. Using results from A179386, it should be possible to prove that the sequence is complete up to a given point. - M. F. Hasler, Sep 26 2013

			The gap 0 appears in 1^3-1^2 or 4^3-8^2 etc.
The gap 2 appears for example in 3^3-5^2.
The gap 4 appears for example in 2^3-2^2 or 5^3-11^2.
The gap 19 appears in 7^3-18^2, the gap 20 in 6^3-14^2.

Essentially the same as A087285.

lst={};Do[a=n^3-Floor[Sqrt[n^3]]^2;If[a<=508,AppendTo[lst,a]],{n,2*8!}]; Take[Union@lst,90]

Edited by R. J. Mathar, Oct 09 2009

Name corrected by M. F. Hasler, Oct 05 2013

cp's OEIS Frontend

A154101 Numbers n with property that A077116(n) is nonzero square.

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A154100 Numbers n with property that n^2 are nonzero square terms in A077116.

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A179388 Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

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A065733 Largest square <= n^3.

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A075847 Difference between n^2 and the largest cube <= n^2.

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A087285 Possible differences between a cube and the next smaller square.

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A077118 Nearest integer square to n^3.

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A077119 a(n) = A077118(n) - n^3.

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