A001235 -id:A001235 - cp's OEIS frontend

A245671 3*x + 1 sequence starting at 1729.

1729, 5188, 2594, 1297, 3892, 1946, 973, 2920, 1460, 730, 365, 1096, 548, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638

Offset: 0

Vincenzo Librandi, Jul 29 2014

1729 is known as the Hardy-Ramanujan number.
This sequence includes all those starting from the odd numbers: 5, 23, 35, 53, 61, 103, 137, 155, 167, 175, 233, 251, 263, 283, 319, 325, 365, 377, 395, 425, 433, 445, 479, 577, 593, 719, 911, 973, 1079, 1297, 1367, 1619.
Reaches a power of 2 at a(100) = 16. - Alonso del Arte, May 30 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to 3x+1 (or Collatz) problem

Row 1729 of A347270.
Cf. A001235.
3*x+1 sequence starting at k: A008884 (k = 27), A008880 (k = 33), A008878 (k = 39), A008883 (k = 51), A008877 (k = 57), A008874 (k = 63), A008876 (k = 81), A008879 (k = 87), A008875 (k = 95), A008873 (k = 97), A008882 (k = 99), this sequence (k = 1729).

[n eq 1 select 1729 else IsOdd(Self(n-1)) select 3*Self(n-1)+1 else Self(n-1) div 2: n in [1..100]];

NestList[If[EvenQ[#], #/2, 3# + 1] &, 1729, 100]

a(0) = 1729, a(n) = 3*a(n-1) + 1 if a(n-1) is odd, a(n) = a(n-1)/2 otherwise. - Alois P. Heinz, Jul 29 2014

A255366 Total number of ON cells at stage n of two-dimensional cellular automaton defined by the rules of the "Ulam-Warburton" two-dimensional cellular automaton (A147562) for two of its wedges and defined by "Rule 750" using the von Neumann neighborhood (A169707) for the two other wedges.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 53, 85, 89, 101, 117, 149, 165, 205, 257, 341, 345, 357, 373, 405, 421, 461, 513, 597, 613, 653, 705, 797, 857, 989, 1141, 1365, 1369, 1381, 1397, 1429, 1445, 1485, 1537, 1621, 1637, 1677, 1729, 1821, 1881, 2013, 2165, 2389, 2405, 2445, 2497

Offset: 1

Omar E. Pol, Feb 21 2015

First differs from A162795 at a(14), but it appears that then they share infinitely many terms. It appears that this is very close to A162795 rather than both A147562 and A169707.
The graphs of both A162795 and this sequence are intertwined.
Note that there are four main versions of this cellular automaton, depending on whether the wedges with the same rule are opposite or perpendicular and also depending on whether each mentioned version is represented by the "one-step rook" illustration or by the "one-step bishop" illustration. The four versions are represented by this sequence.
a(43) = 1729 is also the Hardy-Ramanujan number.

			a(43) = (1705 + 1753)/2 = 3458/2 = 1729.

Cf. A001235, A048645, A139250, A147562, A160164, A162795, A169707.

a(n) = (A147562(n) + A169707(n))/2.

It appears that a(n) = A147562(n) = A162795(n) = A169709(n), if n is a member of A048645, or in other words: if the binary weight of n is 1 or 2, but note that a(n) = A162795(n) for many other values of n.

A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

Original entry on oeis.org

0, 1, 9, 21, 49, 61, 97, 157, 225, 237, 273, 333, 417, 525, 657, 813, 961, 973, 1009, 1069, 1153, 1261, 1393, 1549, 1729, 1933, 2161, 2413, 2689, 2989, 3313, 3661, 3969, 3981, 4017, 4077, 4161, 4269, 4401, 4557, 4737, 4941, 5169, 5421, 5697, 5997, 6321, 6669, 7041, 7437, 7857, 8301, 8769, 9261, 9777, 10317, 10881, 11469

Offset: 0

Omar E. Pol, Apr 21 2015

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
At stage 1, only one cell is turned ON, so a(1) = 1.
If n is a power of 2 so the structure is a square of side length 2n - 1 that contains (2n-1)^2 ON cells.
The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).
Note that a(24) = 1729 is also the Hardy-Ramanujan number (see A001235).
Has the same rules as A256534 but here a(1) = 1 not 4.
Has a smoother behavior than A160414 with which shares infinitely many terms (see example).
A256531, the first differences, gives the number of cells turned ON at n-th stage.

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  157,  225;
237,  273,  333,  417,  525,  657,  813,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A160414 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _ _ _ _ _   _ _ _ _ _ _  |_| |
.      |_ _| |  _ _ _ _  | |  _ _ _ _  | |_ _|
.          | | |  _ _  | | | |  _ _  | | |
.          | | | |  _|_|_|_|_|_|_  | | | |
.          | | | |_|  _ _   _ _  |_| | | |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _ _| |_|  _  |_| |_ _ _| |
.          |  _ _ _|  _| |_| |_  |_ _ _  |
.          | |  _ _| | |_ _ _| | |_ _  | |
.          | | |  _| |_ _| |_ _| |_  | | |
.          | | | | |_ _ _ _ _ _ _| | | | |
.          | | | |_ _| | | | | |_ _| | | |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.

Paolo Xausa, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000225, A011782, A001235, A060867, A139250, A147562, A160410, A160414, A256531, A256534.

With[{z=7},Join[{0},Flatten[Array[(2^#-1)^2+12Range[0,2^(#-1)-1]^2&,z]]]] (* Generates 2^z terms *) (* Paolo Xausa, Nov 15 2023, after Omar E. Pol *)

For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = (2^i-1)^2 + 3*(2*j)^2: next j: next i

A352758 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) - 2*n + 3)/2 for n > 0.

Original entry on oeis.org

5, 148, 1011, 3746, 10081, 22320, 43343, 76606, 126141, 196556, 293035, 421338, 587801, 799336, 1063431, 1388150, 1782133, 2254596, 2815331, 3474706, 4243665, 5133728, 6156991, 7326126, 8654381, 10155580, 11844123, 13734986, 15843721, 18186456, 20779895, 23641318, 26788581, 30240116, 34014931, 38132610

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers D > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference C - D is odd, C - D = 2*n - 1, and the difference of the positive cubes C^3 - D^3 is equal to centered cube numbers, with C > D > B > 0, and A > 0, A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = A352755(n), B = A352756(n), C = A352757(n), and D = a(n) (this sequence).
There are infinitely many such numbers a(n) = D in this sequence.
Subsequence of A352136 and of A352223.

			a(1) = 5 belongs to the sequence as 6^3 - 5^3 = 3^3 + 4^3 = 91 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 148 belongs to the sequence as 151^3 - 148^3 = 46^3 + 47^3 = 201159 and 151 - 148 = 3 = 2*2 - 1.
a(3) = (3*(2*3 - 1)^2*((2*3 - 1)^2 + 2) - 2*3 + 3)/2 = 1011.
a(4) = 3*a(3) - 3*a(2) + a(1) + 576*2 = 3*1011 - 3*148 + 5 + 576*2  = 3746.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352224, A352225, A352755, A352756, A352757, A352759, A355751, A355752, A355753.

restart; for n to 20 do (1/2)*(3*(2*n - 1)^2*((2*n - 1)^2 + 2) - 2*n + 3); end do;

def A352758(n): return n*(n*(n*(24*n - 48) + 48) - 25) + 6 # Chai Wah Wu, Jul 11 2022

A352757(n)^3 - a(n)^3 = A352756(n)^3 + (A352756(n) + 1)^3 = A352755(n) and A352757(n) - a(n) = 2*n - 1.
a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) - 2*n + 3)/2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 576*(n - 2), with a(1) = 5, a(2) = 148 and a(3) = 1011.
a(n) can be extended for negative n such that a(-n) = a(n+1) + (2n + 1).
G.f.: x*(5 + 123*x + 321*x^2 + 121*x^3 + 6*x^4)/(1 - x)^5. - Stefano Spezia, Apr 08 2022

A003826 Numbers that are the sum of two cubes in at least four ways (primitive solutions).

Original entry on oeis.org

6963472309248, 12625136269928, 21131226514944, 26059452841000, 74213505639000, 95773976104625, 159380205560856, 174396242861568, 300656502205416, 376890885439488, 521932420691227, 573880096718136

Offset: 1

N. J. A. Sloane

nonn

R. K. Guy, Unsolved Problems in Number Theory, D1.

Uwe Hollerbach, Table of n, a(n) for n = 1..664
Uwe Hollerbach, Taxi, Taxi! [Original link, broken]; see also Archive.org backup and local cache [top page only]
E. Rosenstiel et al., The Four Least Solutions ..., Instit. of Mathem. and Its Applic. Bull. Jul 27 (pp. 155-157) 1991
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Diophantine Equation--3rd Powers
David W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

Cf. A011541, A023050, A023051, A001235.

More terms from David W. Wilson, Oct 15 1997

b-file extended by Ray Chandler, Jan 19 2009

A018787 Numbers that are the sum of two positive cubes in at least three ways (all solutions).

Original entry on oeis.org

87539319, 119824488, 143604279, 175959000, 327763000, 700314552, 804360375, 958595904, 1148834232, 1407672000, 1840667192, 1915865217, 2363561613, 2622104000, 3080802816, 3235261176, 3499524728, 3623721192, 3877315533, 4750893000, 5544709352, 5602516416

Offset: 1

David W. Wilson, Aug 15 1996

nonn

J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
R. K. Guy, Unsolved Problems in Number Theory, D1.

Ray Chandler, Table of n, a(n) for n=1..100000
Uwe Hollerbach, Taxi, Taxi! [Original link, broken]
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]

Cf. A001235, A003825, A023051, A025294, A025398, A344804.

a=Sort[Flatten@Table[n^3+m^3,{m,2000},{n,m-1,1,-1}]];f3[l_]:=Module[{t={}},Do[If[l[[n]]==l[[n+2]],AppendTo[t,l[[n]]]],{n,1,Length[l]-2}];t];f3[a] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)

A031980 a(n) is the smallest number >= 1 not occurring earlier and not the sum of cubes of two distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

J. Castillo (arp(AT)cia-g.com) [Broken email address?]

Mihaly Bencze [Beneze], Smarandache recurrence type sequences, Bulletin of pure and applied sciences, Vol. 16E, No. 2, 1997, pp. 231-236.
F. Smarandache, Properties of numbers, ASU Special Collections, 1973.

Klaus Brockhaus, Table of n, a(n) for n = 1..4900
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A024670 (sums of cubes of two distinct positive integers), A001235 (sums of two cubes in more than one way), A141805 (complement).

m:=77; a:=[]; a2:={}; for n in [1..m] do p:=1; u:= a2 join { x: x in a }; while p in u do p:=p+1; end while; if p gt m then break; end if; a2:=a2 join { x^3 + p^3: x in a | x^3 + p^3 le m }; Append(~a,p); end for; print a; // Klaus Brockhaus, Jul 16 2008

A031980 = {1}; Do[ m = Ceiling[(n-1)^(1/3)]; s = Select[ A031980, # <= m &]; ls = Length[s]; sumOfCubes = Union[ Flatten[ Table[ s[[i]]^3 + s[[j]]^3, {i, 1, ls}, {j, i+1, ls}]]]; If[ FreeQ[ sumOfCubes, n], AppendTo[ A031980, n] ], {n, 2, 77}]; A031980 (* Jean-François Alcover, Dec 14 2011 *)

More terms from Larry Reeves (larryr(AT)acm.org), Sep 26 2000

Better definition from Klaus Brockhaus, Jul 16 2008

A051302 Numbers whose square can be expressed as the sum of two positive cubes in more than one way.

Original entry on oeis.org

77976, 223587, 623808, 894348, 1788696, 2105352, 2989441, 4298427, 4672423, 4990464, 5986575, 6036849, 7154784, 8437832, 9747000, 14309568, 16842816, 23915528, 24147396, 24770529, 26745768, 27948375, 34387416, 34634719, 36570744, 37379384, 39923712, 47892600

Offset: 1

Jud McCranie

Observations regarding terms through a(64)=306761364: All are multiples of 7^2, 13^2, and/or 19^2. Other than 2, 3, 5 and 11, their only prime factors are 7, 13, 19, 31, 43, 61, 67, 79, 127, 151, and 181 (each of which exceeds a multiple of 6 by 1). None is a cube or higher power; the ones that are squares are a(7), a(12), a(17), a(19), a(20), a(32), a(33), a(41), a(49), a(55), and a(58). - Jon E. Schoenfield, Oct 08 2006
Many of the terms beyond a(64) have prime factors other than those found in a(1) through a(64); however, each term through a(774) has at most one distinct prime factor p > 5 that does not exceed a multiple of 6 by 1, and p, if such a prime factor exists, has a multiplicity m=3, with only a few exceptions: n=651 and n=713 (where p^m is 11^2), n=346 and n=770 (where p^m is 17^2), n=699 and n=740 (where p^m is 23^2), and n=741 (where p^m is 11^6). - Jon E. Schoenfield, Oct 20 2013
First differs from A145553 at A051302(172)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
This sequence is the union of A145553 and A155961.
This sequence is infinite. If n is a member of this sequence, then n^2 = a^3 + b^3 = c^3 + d^3 where (a, b) and (c, d) are distinct pairs. If n^2 = a^3 + b^3 = c^3 + d^3, then (n*k^3)^2 = n^2*k^6 = k^6*(a^3 + b^3) = k^6*(c^3 + d^3) = (a*k^2)^3 + (b*k^2)^3 = (c*k^2)^3 + (d*k^2)^3. It is obvious that if (a, b) and (c, d) are distinct, then (k^2*a, k^2*b), (k^2*c, k^2*d) are also distinct for all nonzero values of k. So if n is in this sequence, then n*k^3 is in this sequence for all k > 0. - Altug Alkan, May 10 2016

			2989441^2 = 1729^3+20748^3 = 15561^3+17290^3, so 2989441 is in the sequence.

Jon E. Schoenfield and Ray Chandler, Table of n, a(n) for n = 1..774

Cf. A050801, A001235, A011541, A145553, A155961.

(* Warning: this script is only a recomputation of the original b-file of 64 terms from Jon E. Schoenfield, and should not be used to extend the data. *)
max = 310000000; cubeFreeParts = {361, 8281, 33124, 159201, 169309, 221725, 565068, 628849, 917427, 1054729, 2370963, 2989441, 4672423, 8968323, 9402967, 9795747, 34634719};
r[x_] := Reduce[0 < y <= z && x^2 == y^3 + z^3, {y, z}, Integers];
okQ[primes_] := Intersection[{2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 67, 79, 127, 139, 151, 181}, primes] == primes;
crop[n_] := Reap[For[m = 1, True, m++, x = n*m^3; If[x > max, Break[]]; If[okQ[FactorInteger[x][[All, 1]]], If[Head[rx = r[x]] === Or, Print["x = ", x, " ", rx]; Sow[x]];]]][[2, 1]];
A051302 = crop /@ cubeFreeParts // Flatten // Sort (* Jean-François Alcover, Jul 02 2017 *)

T=thueinit('x^3+1, 1);
is(n)=my(v=thue(T, n^2)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ Charles R Greathouse IV, May 10 2016

Definition corrected by Jon E. Schoenfield, Aug 27 2006
More terms from Jon E. Schoenfield, Oct 08 2006
Extended by Ray Chandler, Nov 22 2011

A051387 Numbers whose 4th power is the sum of two positive cubes in a nontrivial way.

Original entry on oeis.org

134, 182, 183, 201, 219, 273, 278, 309, 399, 422, 453, 497, 579, 651, 658, 1036, 1132, 1324, 1464, 1482, 1554, 1608, 1612, 1752, 1842, 1996, 2058, 2184, 2457, 2472, 2476, 2483, 2574, 2634, 2994, 3052, 3192, 3465, 3474, 3618, 3624, 3724, 3858, 3906, 3976

Offset: 1

Jud McCranie

nonn

When x is the sum of 2 positive cubes (A003325) there is a trivial solution.
From Chai Wah Wu, Feb 23 2017: (Start)
2457, 4914, 4977, 8001, 8216, ... are terms that are also in A003325.
10202696, 29791125, 48137544, ... are terms that are also in A001235. (End)

			273^4 = 728^3 + 1729^3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001235, A003325, A051386.

A175362 Number of integer pairs (x,y) satisfying |x|^3 + |y|^3 = n, -n <= x,y <= n.

Original entry on oeis.org

1, 4, 4, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

Cube variant of A004018.
Obviously, a(n) must be 4*k, for k >= 0, n > 0. - Altug Alkan, Apr 09 2016
From Robert Israel, Jan 26 2017: (Start)
a(k^3*n) >= a(n) for k >= 1.
a(n) >= 16 for n in A001235.
a(A011541(n)) >= 8*n. (End)

			a(2) = 4 counts (x,y) = (-1,1), (1,1), (-1,-1) and (1,-1).
a(9) = 8 counts (x,y) = (-2,-1), (-2,1), (-1,-2), (-1,2), (1,-2), (1,2), (2,-1) and (2,1).

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001235, A010057, A011541, A025446, A025455, A025464, A025468, A121980.

N:= 200: # to get a(0)..a(N)
G:= (1+2*add(x^(j^3),j=1..floor(N^(1/3))))^2:
seq(coeff(G,x,j),j=0..N); # Robert Israel, Jan 26 2017

a(n) = if(n==0, 1, 4*sum(k=1, sqrtnint(n, 3), ispower(n - k^3, 3))); \\ Daniel Suteu, Aug 16 2021

G.f.: ( 1 + 2 * Sum_{j>=1} x^(j^3) )^2.
a(n^3) = 4 for n > 0. - Altug Alkan, Apr 09 2016
a(n) = 4*Sum_{k=1..floor(n^(1/3))} A010057(n - k^3), for n > 0. - Daniel Suteu, Aug 15 2021

Invalid claim that belonged to A004018 removed by R. J. Mathar, Apr 24 2010

cp's OEIS Frontend

A245671 3*x + 1 sequence starting at 1729.

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A255366 Total number of ON cells at stage n of two-dimensional cellular automaton defined by the rules of the "Ulam-Warburton" two-dimensional cellular automaton (A147562) for two of its wedges and defined by "Rule 750" using the von Neumann neighborhood (A169707) for the two other wedges.

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A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

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A352758 a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) - 2*n + 3)/2 for n > 0.

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A003826 Numbers that are the sum of two cubes in at least four ways (primitive solutions).

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A018787 Numbers that are the sum of two positive cubes in at least three ways (all solutions).

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Mathematica

A031980 a(n) is the smallest number >= 1 not occurring earlier and not the sum of cubes of two distinct earlier terms.

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Magma

Mathematica

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A051302 Numbers whose square can be expressed as the sum of two positive cubes in more than one way.

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A352758 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) - 2*n + 3)/2 for n > 0.