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A060464 Numbers that are not congruent to 4 or 5 mod 9. 10
0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 84, 87, 88, 89, 90, 91 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Conjecture: n is a sum of three cubes iff n is in this sequence.

As of their 2009 paper, Elsenhans and Jahnel did not know of a sum of three cubes that gives 33 or 42.

The problem with 33 is cracked, see links below: 8866128975287528^3 + (-8778405442862239)^3 + (-2736111468807040)^3 = 33. - Alois P. Heinz, Mar 11 2019

Numbers that are congruent to {0, 1, 2, 3, 6, 7, 8} mod 9. - Wesley Ivan Hurt, Jul 21 2016

Heath-Brown conjectures that n is a sum of three cubes in infinitely many ways iff n is in this sequence (and not at all otherwise). See his paper for a conjectural asymptotic. - Charles R Greathouse IV, Mar 12 2019

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, Section D5.

Cohen H. 2007. Number Theory Volume I: Tools and Diophantine Equations. Springer Verlag p. 380. - Artur Jasinski, Apr 30 2010

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..2000

Andrew R. Booker, Cracking the problem with 33, March 2019

Andrew R. Booker and Brady Haran, 42 is the new 33, Numberphile video (2019)

Tim Browning and Brady Haran, The Uncracked Problem with 33, Numberphile video (2015)

Tim Browning and Brady Haran, 74 is cracked, Numberphile video (2016)

Jean-Louis Colliot-Thélène and Olivier Wittenberg, Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines, Amer. J. Math. 134:5 (2012), pp. 1303-1327.

Andreas-Stephan Elsenhans and Jörg Jahnel, List of solutions of x^3 + y^3 + z^3 = n for n < 1000 neither a cube nor twice a cube

A.-S. Elsenhans, J. Jahnel, New sums of three cubes, Math. Comp. 78 (2009) 1227-1230.

D. R. Heath-Brown, The density of zeros of forms for which weak approximation fails, Mathematics of Computation 59 (1992), pp. 613-623.

Sander G. Huisman, Newer sums of three cubes, arXiv:1604.07746 [math.NT], 2016.

H. Mishima, About n=x^3+y^3+z^3

Wikipedia, Manin obstruction

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

FORMULA

G.f.: x^2*(x^3+x^2+1)*(x^3+x+1) / ( (1+x+x^2+x^3+x^4+x^5+x^6)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, Jul 21 2016: (Start)

a(n) = a(n-1) + a(n-7) - a(n-8) for n>8; a(n) = a(n-7) + 9 for n>7.

a(n) = (63*n - 63 + 2*(n mod 7) + 2*((n+1) mod 7) - 12*((n+2) mod 7) + 2*((n+3) mod 7) + 2*((n+4) mod 7) + 2*((n+5) mod 7) + 2*((n+6) mod 7))/49.

a(7k) = 9k-1, a(7k-1) = 9k-2, a(7k-2) = 9k-3, a(7k-3) = 9k-6, a(7k-4) = 9k-7, a(7k-5) = 9k-8, a(7k-6) = 9k-9. (End)

EXAMPLE

30 belongs to this sequence because it has the partition as sum of 3 cubes 30 = (-283059965)^3 + (-2218888517)^3 + (2220422932)^3. - Artur Jasinski, Apr 30 2010, edited by M. F. Hasler, Nov 10 2015

MAPLE

for n from 0 to 100 do if n mod 9 <> 4 and n mod 9 <> 5 then printf(`%d, `, n) fi:od:

MATHEMATICA

a = {}; Do[If[(Mod[n, 9] == 4) || (Mod[n, 9] == 5), , AppendTo[a, n]], {n, 1, 300}]; a (* Artur Jasinski, Apr 30 2010 *)

PROG

(PARI) n=-1; for (m=0, 4000, if (m%9!=4 && m%9!=5, write("b060464.txt", n++, " ", m)); if (n==2000, break)) \\ Harry J. Smith, Jul 05 2009

(PARI) concat(0, Vec(x^2*(x^3+x^2+1)*(x^3+x+1)/((1+x+x^2+x^3+x^4+x^5+x^6)*(x-1)^2) + O(x^100))) \\ Altug Alkan, Nov 06 2015

(PARI) a(n)=n\7*9+[0, 1, 2, 3, 6, 7, 8][n%7+1] \\ Charles R Greathouse IV, Nov 06 2015

(MAGMA) [n : n in [0..150] | n mod 9 in [0, 1, 2, 3, 6, 7, 8]]; // Wesley Ivan Hurt, Jul 21 2016

(GAP) A060464:=Filtered([0..100], n->n mod 9 <>4 and n mod 9 <>5); # Muniru A Asiru, Feb 17 2018

CROSSREFS

Cf. A060465, A060466, A060467.

A156638 is the complement of this sequence.

Sequence in context: A039189 A039141 A008541 * A039102 A287103 A050023

Adjacent sequences:  A060461 A060462 A060463 * A060465 A060466 A060467

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Apr 10 2001

EXTENSIONS

More terms from James A. Sellers, Apr 11 2001

STATUS

approved

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Last modified April 23 14:15 EDT 2019. Contains 322386 sequences. (Running on oeis4.)