A165288 -id:A165288 - cp's OEIS frontend

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

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

A077116 n^3 - A065733(n).

Original entry on oeis.org

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

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",")))

A229618 Numbers that are the distance between a square and the next larger cube.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 15, 18, 19, 20, 25, 26, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 67, 71, 72, 74, 76, 79, 81, 83, 87, 100, 104, 106, 107, 109, 112, 116, 118, 126, 127, 128, 135, 139, 143

Offset: 1

M. F. Hasler, Sep 26 2013

This is the range of the sequence A181138 (= least k>0 such that n^2+k is a cube). Note that this is not the same as A087285 = range of A077116 = difference between a cube and the next smaller square: 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., 9+18 = 27 = 3^3 is the least cube above 9 = 3^2, but 25 = 5^2 is the largest square below 27. Therefore the number 18 is in this sequence, but not in A087285.
See A077116 and A181138 and A179386 for motivations.
Apart from the leading 1, this is a subsequence of A106265, which does not require the square to be the next smaller one: For example, 23 = 27 - 4 = 3^3 - 2^2 is in A106265 but not in this sequence. A165288 is a subsequence of this one, except for the initial term.

			a(1) = 1 = 1^3-0^2 (but this is the only solution to y^3-x^2 = 1).
a(2) = 2 = 27-25 (= 3^3-5^2), and this is the only solution to y^3-x^2 = 2.
The number 3 is not in the sequence since there are no x, y > 0 such that y^3-x^2 = 3.
a(3) = 4 = 8-4 (= 2^3-2^2) = 125-121 (= 5^3-11^2); these are the only two solutions to y^3-x^2 = 4, for all x>11, the minimal positive y^3-x^2 is 7.

Cf. A087285, A087286, A088017, A081121, A081120, A077116, A065733.

A165289 Nonnegative integers of the form m^2 - floor(m^(2/3))^3 where m is a positive integer.

Original entry on oeis.org

0, 1, 3, 8, 9, 12, 15, 17, 18, 19, 22, 24, 30, 36, 37, 38, 40, 44, 55, 57, 64, 65, 68, 71, 73, 79, 80, 89, 97, 98, 100, 101, 106, 107, 108, 112, 113, 119, 121, 128, 129, 138, 141, 145, 148, 151, 154, 156, 161, 163, 164, 168, 169, 171, 172, 190, 196, 197, 198, 204, 208

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

nonn

The positive terms form a subsequence of A087286. Some terms of A087286 are missing here, the smallest such number being 3781 = (6^3)^2 - (6^2-1)^3. [From Max Alekseyev, Jun 19 2011]

Cf. A087286, A165288.

lst={}; Do[a=(x=n^2)-(y=Floor[(n^2)^(1/3)]^3); If[a<=416,AppendTo[lst,a]], {n,8!}]; Take[Union@lst,100]

Minor edits by N. J. A. Sloane, Oct 24 2009

Definition corrected by Max Alekseyev, Jun 19 2011

A165290 Numbers which cannot be represented as a cube - nearest square (cube >= square).

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 41, 42, 43, 44, 46, 50, 51, 52, 54, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

nonn

Complement of A165288.

Cf. A165288, A165289

lst={}; Do[a=n^3-Floor[Sqrt[n^3]]^2; If[a<=508,AppendTo[lst,a]],{n,2*8!}]; lst=Take[Union@lst,90]; lst1={}; Do[AppendTo[lst1,n],{n,508}]; lst1; Complement[lst1,lst]

A165291 Complement of A165289.

Original entry on oeis.org

2, 4, 5, 6, 7, 10, 11, 13, 14, 16, 20, 21, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 39, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 66, 67, 69, 70, 72, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 102

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

nonn

Numbers which are impossible values for the difference of a square minus the nearest smaller or equal cube.

Cf. A165288, A165289, A165290

lst={};Do[a=(x=n^2)-(y=Floor[(n^2)^(1/3)]^3);If[a<=416,AppendTo[lst,a]], {n,8!}];Take[Union@lst,100]; lst1={};Do[AppendTo[lst1,n],{n,416}]; lst1; Complement[lst1,lst]

Definition simplified - R. J. Mathar, Sep 21 2009

cp's OEIS Frontend

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

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A077116 n^3 - A065733(n).

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

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A229618 Numbers that are the distance between a square and the next larger cube.

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A165289 Nonnegative integers of the form m^2 - floor(m^(2/3))^3 where m is a positive integer.

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A165290 Numbers which cannot be represented as a cube - nearest square (cube >= square).

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A165291 Complement of A165289.

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