Search: a011541 -id:a011541
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A230564
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Rational rank of the n-th taxicab elliptic curve x^3 + y^3 = A011541(n).
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+20
1
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OFFSET
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1,2
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COMMENTS
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Guy, 2004: "Andrew Bremner has computed the rational rank of the elliptic curve x^3 + y^3 = Taxicab(n) as equal to 2, 4, 5, 4 for n = 2, 3, 4, 5, respectively."
Abhinav Kumar computed that a(1) = 0 (see the MathOverflow link for details). But Euler and Legendre scooped him (see the next comment).
Noam D. Elkies: "... the fact that x^3+y^3=2 has no [rational] solutions other than x=y=1 is attributed by Dickson to Euler himself: see Dickson's History of the Theory of Numbers (1920) Vol.II, Chapter XXI "Numbers the Sum of Two Rational Cubes", page 572. The reference (footnote 182) is "Algebra, 2, 170, Art. 247; French transl., 2, 1774, pp. 355-60; Opera Omnia, (1), I, 491". In the next page Dickson also refers to work of Legendre that includes this result (footnote 184: "Théorie des nombres, Paris, 1798, 409; ...")." See the MathOverflow link for further comments from Elkies.
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, D1.
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LINKS
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FORMULA
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EXAMPLE
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rank(x^3 + y^3 = 2) = 0.
rank(x^3 + y^3 = 1729) = 2.
rank(x^3 + y^3 = 87539319) = 4.
rank(x^3 + y^3 = 6963472309248) = 5.
rank(x^3 + y^3 = 48988659276962496) = 4.
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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STATUS
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approved
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OFFSET
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1,1
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LINKS
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KEYWORD
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base,nonn,hard
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AUTHOR
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STATUS
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approved
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2, 1, 7, 2, 9, 8, 7, 5, 3, 9, 3, 1, 9, 6, 9, 6, 3, 4, 7, 2, 3, 0, 9, 2, 4, 8, 4, 8, 9, 8, 8, 6, 5, 9, 2, 7, 6, 9, 6, 2, 4, 9, 6
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OFFSET
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1,1
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LINKS
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KEYWORD
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base,easy,nonn
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AUTHOR
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STATUS
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approved
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OFFSET
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1,1
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COMMENTS
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If b is the sum of two positive cubes in exactly n ways, then b*c^3 is the sum of two positive cubes in at least n ways for all c > 0. So A011541(i)*c^3 is a candidate for unknown Taxi-cab numbers. a(6) = 79 is an interesting example that is related to this case. Additionally, the benefit of this simple fact is the determination of upper bounds for unknown Taxi-cab numbers in relatively easy way.
The inequalities that are given in the comment section of A011541 are:
So a(6) <= 101, a(7) <= 127, a(8) <= 139.
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LINKS
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EXAMPLE
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a(5) = 79 because abs(A011541(6) - A011541(5)*79^3) = abs(24153319581254312065344 - 48988659276962496*79^3) = 0
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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A003325
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Numbers that are the sum of 2 positive cubes.
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+10
136
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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
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OFFSET
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1,1
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COMMENTS
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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]
(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
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REFERENCES
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C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 354.
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LINKS
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) 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
(PARI) 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
(PARI) 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
(PARI) 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
(Haskell)
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
(Python)
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]
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Error in formula line corrected by Zak Seidov, Jul 23 2009
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STATUS
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approved
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A001235
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Taxi-cab numbers: sums of 2 cubes in more than 1 way.
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+10
114
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1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597, 439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, 684019, 704977
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OFFSET
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1,1
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COMMENTS
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From Wikipedia: "1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 'I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."'"
A011541 gives another version of "taxicab numbers".
If n is in this sequence, then n*k^3 is also in this sequence for all k > 0. So this sequence is obviously infinite. - Altug Alkan, May 09 2016
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, Section D1.
G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.
Ya. I. Perelman, Algebra can be fun, pp. 142-143.
H. W. Richmond, On integers which satisfy the equation t^3 +- x^3 +- y^3 +- z^3, Trans. Camb. Phil. Soc., 22 (1920), 389-403, see p. 402.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165.
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LINKS
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A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
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EXAMPLE
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4104 belongs to the sequence as 4104 = 2^3 + 16^3 = 9^3 + 15^3.
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MATHEMATICA
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Select[Range[750000], Length[PowersRepresentations[#, 2, 3]]>1&] (* Harvey P. Dale, Nov 25 2014, with correction by Zak Seidov, Jul 13 2015 *)
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PROG
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(PARI) is(n)=my(t); for(k=ceil((n/2)^(1/3)), (n-.4)^(1/3), if(ispower(n-k^3, 3), if(t, return(1), t=1))); 0 \\ Charles R Greathouse IV, Jul 15 2011
(PARI) T=thueinit(x^3+1, 1);
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CROSSREFS
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Solutions in greater numbers of ways:
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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A272885
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Cubefree taxi-cab numbers.
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+10
20
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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
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OFFSET
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1,1
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COMMENTS
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Taxi-cab numbers (A001235) that are not divisible by any cube > 1.
Cubeful taxi-cab numbers are 4104, 13832, 32832, 39312, 46683, 64232, 65728, 110656, 110808, 134379, 165464, 171288, ...
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LINKS
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EXAMPLE
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195841 is a term because 195841 is a member of A001235 and 195841 = 37*67*79.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A160414
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Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).
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+10
19
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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;
...
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.
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Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
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. | | _|_|_ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ |_| |
. |_ _| | _|_ _|_ | | _|_ _|_ | |_ _|
. | |_| _ _ |_| |_| _ _ |_| |
. | | | _|_|_ _ _|_|_ | | |
. | _| |_| _ _ _ _ |_| |_ |
. | | |_ _| | _|_|_ | |_ _| | |
. | |_ _| | |_| _ |_| | |_ _| |
. | _ _ | _| |_| |_ | _ _ |
. | | _|_| | |_ _ _| | |_|_ | |
. | |_| _| |_ _| |_ _| |_ |_| |
. | | | |_ _ _ _ _ _ _| | | |
. | _| |_ _| |_ _| |_ _| |_ |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _| |_ _| |_ _| |_ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
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After 10 generations there are 273 ON cells, so a(10) = 273.
(End)
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(PARI) my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<<valuation(n, 2)), n\2)); vector(99, i, 4*(s+=t(i))+1) \\ Altug Alkan, Sep 25 2015
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CROSSREFS
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Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A083737
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Pseudoprimes to bases 2, 3 and 5.
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+10
11
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1729, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 670033, 721801, 748657
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OFFSET
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1,1
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COMMENTS
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a(n) = n-th positive integer k(>1) such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k)
See A153580 for numbers k > 1 such that 2^k-2, 3^k-3 and 5^k-5 are all divisible by k but k is not a Carmichael number (A002997).
Note that a(1)=1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jan 18 2009
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LINKS
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EXAMPLE
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a(1)=1729 since it is the first number such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k).
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MATHEMATICA
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Select[ Range[838200], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 & ]
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PROG
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(PARI) is(n)=!isprime(n)&&Mod(2, n)^(n-1)==1&&Mod(3, n)^(n-1)==1&&Mod(5, n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003
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EXTENSIONS
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STATUS
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approved
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A023051
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Numbers that are the sum of two positive cubes in at least four ways (all solutions).
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+10
9
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6963472309248, 12625136269928, 21131226514944, 26059452841000, 55707778473984, 74213505639000, 95773976104625, 101001090159424, 159380205560856, 169049812119552, 174396242861568, 188013752349696
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OFFSET
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1,1
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, D1.
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LINKS
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Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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