A001235 -id:A001235 - cp's OEIS frontend

A274245 Numbers n such that n*(n+1)/2 is a Taxi-cab number (A001235).

349999, 591408, 405332018, 525796270

Offset: 1

Altug Alkan, Jul 02 2016

In other words, numbers n such that 0 + 1 + 2 + ... + n = a^3 + b^3 = c^3 + d^3 where (a, b) and (c, d) are distinct pairs and a, b, c, d > 0 is soluble.
It is known that there is no triangular number that is also a cube except 0 and 1. So if the sum of k positive cubes is a triangular number that is bigger than 1, then the minimum value of k is 2. At this point sequence focuses on that question: What are the triangular numbers that are the sum of two positive cubes in more than one way?
A000217(349999) = 61249825000 is the least triangular number that is also a Taxi-cab number.
a(5) > 10^9. - Giovanni Resta, Jul 04 2016

			349999 is a term because 349999*(349999+1) / 2 = 61249825000 = 820^3 + 3930^3 = 3018^3 + 3232^3.
591408 is a term because 591408*(591408+1) / 2 = 174882006936 = 2070^3 + 5496^3 = 3238^3 + 5204^3.

Cf. A000217, A001235.

a(3)-a(4) from Giovanni Resta, Jul 04 2016

A274367 Taxi-cab numbers (A001235) that are of the form x^2 + y^4 in more than one way (x, y > 0).

Original entry on oeis.org

27445392, 1644443281, 2367885312, 5687433577, 112416325632, 208265121792, 900069054976, 1976398601697, 6735639678976, 9698858237952, 9911785815477, 14585606569872, 15283760730112, 18156501172017, 23295727931392, 29871321586561, 33510832422912, 67250060669952

Offset: 1

Altug Alkan, Jun 19 2016

nonn

A272701(3) = 27445392 is the least number with the property that sequence focuses on.

If n = a^3 + b^3 = c^3 + d^3 = x^2 + y^4 = z^2 + t^4, then n*k^12 = (a*k^4)^3 + (b*k^4)^3 = (c*k^4)^3 + (d*k^4)^3 = (x*k^6)^2 + (y*k^3)^4 = (z*k^6)^2 + (t*k^3)^4. So if n is this sequence, then n*k^12 is also in this sequence for all k > 1.

			27445392 is a term because 27445392 = 141^3 + 291^3 = 198^3 + 270^3 = 756^2 + 72^4 = 5076^2 + 36^4.
112416325632 is a term because 112416325632 = 27445392*2^12.

Cf. A001235, A111925, A272701.

a(2)-a(18) from Giovanni Resta, Jun 19 2016

A003325 Numbers that are the sum of 2 positive cubes.

Original entry on oeis.org

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343

Offset: 1

N. J. A. Sloane

It is conjectured that this sequence and A052276 have infinitely many numbers in common, although only one example (128) is known. [Any further examples are greater than 5 million. - Charles R Greathouse IV, Apr 12 2020] [Any further example is greater than 10^12. - M. F. Hasler, Jan 10 2021]
A113958 is a subsequence; if m is a term then m+k^3 is a term of A003072 for all k > 0. - Reinhard Zumkeller, Jun 03 2006
From James R. Buddenhagen, Oct 16 2008: (Start)
(i) N and N+1 are both the sum of two positive cubes if N=2*(2*n^2 + 4*n + 1)*(4*n^4 + 16*n^3 + 23*n^2 + 14*n + 4), n=1,2,....
(ii) For n >= 2, let N = 16*n^6 - 12*n^4 + 6*n^2 - 2, so N+1 = 16*n^6 - 12*n^4 + 6*n^2 - 1.
Then the identities 16*n^6 - 12*n^4 + 6*n^2 - 2 = (2*n^2 - n - 1)^3 + (2*n^2 + n - 1)^3 16*n^6 - 12*n^4 + 6*n^2 - 1 = (2*n^2)^3 + (2*n^2 - 1)^3 show that N, N+1 are in the sequence. (End)
If n is a term then n*m^3 (m >= 2) is also a term, e.g., 2m^3, 9m^3, 28m^3, and 35m^3 are all terms of the sequence. "Primitive" terms (not of the form n*m^3 with n = some previous term of the sequence and m >= 2) are 2, 9, 28, 35, 65, 91, 126, etc. - Zak Seidov, Oct 12 2011
This is an infinite sequence in which the first term is prime but thereafter all terms are composite. - Ant King, May 09 2013
By Fermat's Last Theorem (the special case for exponent 3, proved by Euler, is sufficient), this sequence contains no cubes. - Charles R Greathouse IV, Apr 03 2021

C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 354.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
C. G. J. Jacobi, Gesammelte Werke.
Michael Penn, 1674 is not a perfect cube, 2020 video
N. J. A. Sloane, Table of n, a(n) for n = 1..59562
D. Tournes, A Glance on Indian Mathematician Srinivasa Ramanujan(1887-1920). [Text in French]
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Subsequence of A004999 and hence of A045980; supersequence of A202679.

Cf. A024670 (2 distinct cubes), A003072, A001235, A011541, A003826, A010057, A000578, A027750, A010052, A085323 (n such that a(n+1)=a(n)+1).

a003325 n = a003325_list !! (n-1)
a003325_list = filter c2 [1..] where
   c2 x = any (== 1) $ map (a010057 . fromInteger) $
                       takeWhile (> 0) $ map (x -) $ tail a000578_list
-- Reinhard Zumkeller, Mar 24 2012

nn = 2*20^3; Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]] (* T. D. Noe, Oct 12 2011 *)
With[{upto=2000},Select[Total/@Tuples[Range[Ceiling[Surd[upto,3]]]^3,2],#<=upto&]]//Union (* Harvey P. Dale, Jun 11 2016 *)

cubes=sum(n=1, 11, x^(n^3), O(x^1400)); v = select(x->x, Vec(cubes^2), 1); vector(#v, k, v[k]+1) \\ edited by Michel Marcus, May 08 2017

isA003325(n) = for(k=1,sqrtnint(n\2,3), ispower(n-k^3,3) && return(1)) \\ M. F. Hasler, Oct 17 2008, improved upon suggestion of Altug Alkan and Michel Marcus, Feb 16 2016

T=thueinit('z^3+1); is(n)=#select(v->min(v[1],v[2])>0, thue(T,n))>0 \\ Charles R Greathouse IV, Nov 29 2014

list(lim)=my(v=List()); lim\=1; for(x=1,sqrtnint(lim-1,3), my(x3=x^3); for(y=1,min(sqrtnint(lim-x3,3),x), listput(v, x3+y^3))); Set(v) \\ Charles R Greathouse IV, Jan 11 2022

from sympy import integer_nthroot
def aupto(lim):
  cubes = [i*i*i for i in range(1, integer_nthroot(lim-1, 3)[0] + 1)]
  sum_cubes = sorted([a+b for i, a in enumerate(cubes) for b in cubes[i:]])
  return [s for s in sum_cubes if s <= lim]
print(aupto(1343)) # Michael S. Branicky, Feb 09 2021

Error in formula line corrected by Zak Seidov, Jul 23 2009

A011541 Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.

Original entry on oeis.org

2, 1729, 87539319, 6963472309248, 48988659276962496, 24153319581254312065344

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The sequence is infinite: Fermat proved that numbers expressible as a sum of two positive integral cubes in n different ways exist for any n. Hardy and Wright give a proof in Theorem 412 of An Introduction of Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition).
A001235 gives another definition of "taxicab numbers".
David W. Wilson reports a(6) <= 8230545258248091551205888. [But see next line!]
Randall L Rathbun has shown that a(6) <= 24153319581254312065344.
C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?, 2003, show that with high probability, a(6) = 24153319581254312065344.
When negative cubes are allowed, such terms are called "Cabtaxi" numbers, cf. Boyer's web page, Wikipedia or MathWorld. - M. F. Hasler, Feb 05 2013
a(7) <= 24885189317885898975235988544. - Robert G. Wilson v, Nov 18 2012
a(8) <= 50974398750539071400590819921724352 = 58360453256^3 + 370298338396^3 = 7467391974^3 + 370779904362^3 = 39304147071^3 + 370633638081^3 = 109276817387^3 + 367589585749^3 = 208029158236^3 + 347524579016^3 = 224376246192^3 + 341075727804^3 = 234604829494^3 + 336379942682^3 = 288873662876^3 + 299512063576^3. - PoChi Su, May 16 2013
a(9) <= 136897813798023990395783317207361432493888. - PoChi Su, May 17 2013
From PoChi Su, Oct 09 2014: (Start)
The preceding bounds are not the best that are presently known.
An upper bound for a(22) was given by C. Boyer (see the C. Boyer link), namely
BTa(22)= 2^12 *3^9 * 5^9 *7^4 *11^3 *13^6 *17^3 *19^3 *31^4 *37^4 *43 *61^3 *73 *79^3 *97^3 *103^3 *109^3 *127^3 *139^3 *157 *181^3 *197^3 *397^3 *457^3 *503^3 *521^3 *607^3 *4261^3.
We also know that (97*491)^3*BTa(22) is an upper bound on a(23), corresponding to the sum x^3+y^3 with
x=2^5 *3^4 *5^3 *7 *11 *13^2 *17 *19^2 *31 *37 *61 *79 *103 *109 *127 *139 *181 *197 *397 *457 *503 *521 *607 *4261 *11836681,
y=2^4 *3^3 *5^3 *7 *11 *13^2 *17 *19 *31 *37 *61 *79 *89 *103 *109 *127 *139 *181 *197 *397 * 457 *503 * 521 *607 *4261 *81929041.
(End)
Conjecture: the number of distinct prime factors of a(n) is strictly increasing as n grows (this is not true if a(7) is equal to the upper bound given above), but never exceeds 2*n. - Sergey Pavlov, Mar 01 2017

			From _Zak Seidov_, Mar 22 2013: (Start)
Values of {b,c}, a(n) = b^3 + c^3:
n = 1: {1,1}
n = 2: {1, 12}, {9, 10}
n = 3: {167, 436}, {228, 423}, {255, 414}
n = 4: {2421, 19083}, {5436, 18948}, {10200, 18072}, {13322, 16630}
n = 5: {38787, 365757}, {107839, 362753}, {205292, 342952}, {221424, 336588}, {231518, 331954}
n = 6: {582162, 28906206}, {3064173, 28894803}, {8519281, 28657487}, {16218068, 27093208}, {17492496, 26590452}, {18289922, 26224366}. (End)

C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
R. K. Guy, Unsolved Problems in Number Theory, D1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition), see Theorem 412.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.

D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
Christian Boyer, New upper bounds on Taxicab and Cabtaxi numbers
Christian Boyer, New upper bounds for Taxicab and Cabtaxi numbers, JIS 11 (2008) 08.1.6.
"Durango" Bill Butler, Durango Bill's Ramanujan Numbers and The Taxicab Problem
Cristian S. Calude, Elena Calude and Michael J. Dinneen, What is the value of Taxicab(6)?
Cristian S. Calude, Elena Calude and Michael J. Dinneen, What is the value of Taxicab(6)?, J. Universal Computer Science, 9 (2003), 1196-1203.
Pavel Emelyanov, On Hunting for Taxicab Numbers, arXiv:0802.1147 [math.NT], 2008.
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
U. Hollerbach, The sixth taxicab number is 24153319581254312065344, posting to the NMBRTHRY mailing list, Mar 09 2008.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Dave McKee, Taxicab numbers, Apr 24 2001.
Jean-Charles Meyrignac, The Taxicab Problem
Ken Ono and Sarah Trebat-Leder, The 1729 K3 surface, arXiv:1510.00735 [math.NT], 2015.
Ivars Peterson, Math Trek, Taxicab Numbers
Randall L. Rathbun, Sixth Taxicab Number?, posting to the NMBRTHRY mailing list, Jul 16 2002.
Walter Schneider, Taxicab Numbers
Joseph H. Silverman, Taxicabs and Sums of Two Cubes, American Mathematical Monthly, Volume 100, Issue 4 (Apr., 1993), 331-340.
Po-Chi Su, More Upper Bounds on Taxicab and Cabtaxi Numbers, Journal of Integer Sequences, 19 (2016), #16.4.3.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Taxicab Number
Wikipedia, Taxicab number
D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.
D. W. Wilson, Taxicab Numbers (last snapshot available on web.archive.org, as of June 2013).

Cf. A001235, A003826, A023050, A047696, A080642 (cubefree taxicab numbers).

a(n) <= A080642(n) for n > 0, with equality for n = 1, 2 (only?). - Jonathan Sondow, Oct 25 2013

a(n) > 113*n^3 for n > 1 (a trivial bound based on the number of available cubes; 113 < (1 - 2^(-1/3))^(-3)). - Charles R Greathouse IV, Jun 18 2024

Added a(6), confirmed by Uwe Hollerbach, communicated by Christian Schroeder, Mar 09 2008

A324316 Primary Carmichael numbers.

Original entry on oeis.org

1729, 2821, 29341, 46657, 252601, 294409, 399001, 488881, 512461, 1152271, 1193221, 1857241, 3828001, 4335241, 5968873, 6189121, 6733693, 6868261, 7519441, 10024561, 10267951, 10606681, 14469841, 14676481, 15247621, 15829633, 17098369, 17236801, 17316001, 19384289, 23382529, 29111881, 31405501, 34657141, 35703361, 37964809

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 21 2019

Squarefree integers m > 1 such that if prime p divides m, then the sum of the base-p digits of m equals p. It follows that m is then a Carmichael number (A002997).
Dickson's conjecture implies that the sequence is infinite, see Kellner 2019.
If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(66337/132673) = 0.7071..., where the bound is sharp.
The distribution of primary Carmichael numbers is A324317.
See Kellner and Sondow 2019 and Kellner 2019.
Primary Carmichael numbers are special polygonal numbers A324973. The rank of the n-th primary Carmichael number is A324976(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 26 2019
The first term is the Hardy-Ramanujan number. - Omar E. Pol, Jan 09 2020

			1729 = 7 * 13 * 19 is squarefree, and 1729 in base 7 is 5020_7 = 5 * 7^3 + 0 * 7^2 + 2 * 7 + 0 with 5+0+2+0 = 7, and 1729 in base 13 is a30_13 with a+3+0 = 10+3+0 = 13, and 1729 in base 19 is 4f0_19 with 4+f+0 = 4+15+0 = 19, so 1729 is a member.

Bernd C. Kellner, Table of n, a(n) for n = 1..10000 (computed by using Pinch's database, see link below)
Bernd C. Kellner, On primary Carmichael numbers, #A38 Integers 22 (2022), 39 p.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 p.; arXiv:1902.10672 [math.NT], 2019.
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
Index entries for sequences related to Carmichael numbers.

Subsequence of A002997, A324315.
Least primary Carmichael number with n prime factors is A306657.
Cf. also A005117, A195441, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405, A324973, A324976, A001235.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
Select[Range[1, 10^7, 2], TestCP[#] &]

use ntheory ":all"; my $m; forsquarefree { $m=$; say if @ > 2 && is_carmichael($m) && vecall { $ == vecsum(todigits($m,$)) } @; } 1e7; # _Dana Jacobsen, Mar 28 2019

from sympy import factorint
from sympy.ntheory import digits
def ok(n):
    pf = factorint(n)
    if n < 2 or max(pf.values()) > 1: return False
    return all(sum(digits(n, p)[1:]) == p for p in pf)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 03 2022

a_1 + a_2 + ... + a_k = p if p is prime and m = a_1 * p + a_2 * p^2 + ... + a_k * p^k with 0 <= a_i <= p-1 for i = 1, 2, ..., k (note that a_0 = 0).

A050791 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Sequence gives values of z in monotonic increasing order.

Original entry on oeis.org

12, 103, 150, 249, 495, 738, 1544, 1852, 1988, 2316, 4184, 5262, 5640, 8657, 9791, 9953, 11682, 14258, 21279, 21630, 31615, 36620, 36888, 38599, 38823, 40362, 41485, 47584, 57978, 59076, 63086, 73967, 79273, 83711, 83802, 86166, 90030

Offset: 1

Patrick De Geest, Sep 15 1999

Numbers n such that n^3+1 is expressible as the sum of two nonzero cubes (both greater than 1).
Values of z associated with A050794.
Sequence is infinite. One subsequence is (from x = 1 + 9 m^3, y = 9 m^4, z = 3*m*(3*m^3 + 1), x^3 + y^3 = z^3 + 1): z(m) = 3*m*(3*m^3 + 1) = {12, 150, 738, 2316, 5640, 11682, 21630, 36888, 59076, 90030, ...} = a (1, 3, 6, 10, 13, 17, 20, 23, 30, 37, ...). - Zak Seidov, Sep 16 2013
Numbers n such that n^3+1 is a member of A001235. - Altug Alkan, May 09 2016

			12 is a term because 10^3 + 9^3 = 12^3 + 1 (= 1729).
2316 is in the sequence because 577^3 + 2304^3 = 2316^3 + 1.

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Lewis Mammel, Table of n, a(n) for n = 1..368
Noam Elkies, Rational points near curves and small nonzero |x^3-y^2| via lattice reduction, arXiv:math/0005139 [math.NT], 2000.
S. Ramanujan, Question 681, J. Ind. Math. Soc.
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050792, A050793, A050794, A050787, A229383.

r[z_] := Reduce[ 1 < x < y < z && x^3 + y^3 == z^3 + 1, {x, y}, Integers]; z = 4; A050791 = {}; While[z < 10^4, If[r[z] =!= False, Print[z]; AppendTo[A050791, z]]; z++]; A050791 (* Jean-François Alcover, Dec 27 2011 *)

is(n)=if(n<2,return(0));my(c3=n^3);for(a=2,sqrtnint(c3-5,3),if(ispower(c3-1-a^3,3),return(1)));0 \\ Charles R Greathouse IV, Oct 26 2014

T=thueinit('x^3+1); is(n)=n>8&&#select(v->min(v[1], v[2])>1, thue(T, n^3+1))>0 \\ Charles R Greathouse IV, Oct 26 2014

More terms from Michel ten Voorde
Extended through 47584 by Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Edited by N. J. A. Sloane, May 08 2007

A272885 Cubefree taxi-cab numbers.

Original entry on oeis.org

1729, 20683, 40033, 149389, 195841, 327763, 443889, 684019, 704977, 1845649, 2048391, 2418271, 2691451, 3242197, 3375001, 4342914, 4931101, 5318677, 5772403, 5799339, 6058747, 7620661, 8872487, 9443761, 10702783, 10765603, 13623913, 16387189, 16776487, 16983854, 17045567, 18406603

Offset: 1

Altug Alkan, May 08 2016

nonn

Taxi-cab numbers (A001235) that are not divisible by any cube > 1.
A080642 belongs to another version that A011541 focuses on.
Cubeful taxi-cab numbers are 4104, 13832, 32832, 39312, 46683, 64232, 65728, 110656, 110808, 134379, 165464, 171288, ...

			195841 is a term because 195841 is a member of A001235 and 195841 = 37*67*79.

Cf. A001235, A004709, A011541, A080642.

A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).

Original entry on oeis.org

0, 1, 9, 21, 49, 61, 97, 133, 225, 237, 273, 309, 417, 453, 561, 669, 961, 973, 1009, 1045, 1153, 1189, 1297, 1405, 1729, 1765, 1873, 1981, 2305, 2413, 2737, 3061, 3969, 3981, 4017, 4053, 4161, 4197, 4305, 4413, 4737, 4773, 4881, 4989, 5313, 5421, 5745

Offset: 0

Omar E. Pol, May 20 2009

The structure has a fractal behavior similar to the toothpick sequence A139250.
First differences: A161415, where there is an explicit formula for the n-th term.
For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.

			From _Omar E. Pol_, Sep 24 2015: (Start)
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,  133,  225;
237,  273,  309,  417,  453, 561,  669,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A256530 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.
(End)

Paolo Xausa, Table of n, a(n) for n = 0..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the structure after 24th stage (contains 1729 ON cells)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.

read("transforms") ; isA000079 := proc(n) if type(n,'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4*A048883(n-1)-2*n ; else 4*A048883(n-1) ; end if; end proc:
A160414 := proc(n) add( A161415(k),k=1..n) ; end proc: seq(A160414(n),n=0..90) ; # R. J. Mathar, Oct 16 2010

A160414list[nmax_]:=Accumulate[Table[If[n<2,n,4*3^DigitCount[n-1,2,1]-If[IntegerQ[Log2[n]],2n,0]],{n,0,nmax}]];A160414list[100] (* Paolo Xausa, Sep 01 2023, after R. J. Mathar *)

my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<Altug Alkan, Sep 25 2015

a(n) = 1 + 4*A219954(n), n >= 1. - M. F. Hasler, Dec 02 2012

a(2^k) = (2^(k+1) - 1)^2. - Omar E. Pol, Jan 05 2013

Edited by N. J. A. Sloane, Jun 15 2009 and Jul 13 2009

More terms from R. J. Mathar, Oct 16 2010

cp's OEIS Frontend

A062924 Probably an incomplete version of A001235.

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A273186 Taxi-cab numbers (A001235) n such that n+1 is the sum of two positive cubes (A003325).

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A274245 Numbers n such that n*(n+1)/2 is a Taxi-cab number (A001235).

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A274367 Taxi-cab numbers (A001235) that are of the form x^2 + y^4 in more than one way (x, y > 0).

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A003325 Numbers that are the sum of 2 positive cubes.

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A011541 Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.

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A324316 Primary Carmichael numbers.

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A050791 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Sequence gives values of z in monotonic increasing order.

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