| Displaying 1-5 of 5 results found.
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9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 513, 520, 539, 559, 576, 637, 728, 855, 1001, 1008, 1027, 1064, 1125, 1216, 1332, 1339, 1343, 1358, 1395, 1456, 1512, 1547, 1674, 1729, 1736, 1755, 1792, 1843, 1853
COMMENTS
Not a supersequence of A001235; 7094269 is the smallest number that is in A001235 but not in this sequence (see third example below), the next number is 11261376.
EXAMPLE
9 is the sum of two distinct nonzero cubes in exactly one way: 9 = 1^3 + 2^3. 9 is not in A031980 because 1 and 2 are earlier terms of A031980. Therefore 9 is a term of this sequence. 1729 is the sum of two distinct nonzero cubes in exactly two ways: 1729 = 9^3 + 10^3 = 1^3 + 12^3. 1729 is not in A031980 because 1 and 12 are earlier terms of A031980. Therefore 1729 is a term of this sequence. 7094269 is the sum of two distinct nonzero cubes in exactly two ways: 7094269 = 70^3 + 189^3 = 133^3 + 168^3. 7094269 is in A031980 because it not the sum of cubes of two earlier terms of A031980; in the first case 189 and in the second case 133 is not a term of A031980. Therefore 7094269 is not a term of this sequence.
MATHEMATICA
max = 2000; 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, max}]; Complement[Range[max], A031980] (* Jean-François Alcover, Sep 03 2013 *)
PROG
(Magma) m:=1853; 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 a2;
CROSSREFS
Cf. A141806, A031980 (smallest number not occurring earlier and not the sum of cubes of two distinct earlier terms), A024670 (sums of cubes of two distinct positive integers), A001235 (sums of two cubes in more than one way).
Indices k such that A292547(k) = 0.
+10
3
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 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, 65, 66, 67, 68, 69, 70
COMMENTS
Conjecture: for k > 212594 there are no more terms in this sequence (tested for k < 63000000).
EXAMPLE
3 is in the sequence because A292547(3) = 0 8 is not in the sequence because A292547(8) = -1 201254 is in the sequence because A292547(201254) = 0 212594 is in the sequence because A292547(212594) = 0
MATHEMATICA
With[{nn = 200}, -1 + Position[#, 0][[All, 1]] &@ CoefficientList[ Series[Product[1 + x^((2 k - 1)^3), {k, 1, Floor[nn^(1/3)/2] + 1}], {x, 0, nn}], x]] (* Michael De Vlieger, Sep 22 2017, after Vaclav Kotesovec at A292547 *)
730, 737, 756, 793, 854, 945, 1072, 1241, 2060, 2457, 2926, 3473, 4825, 5642, 6561, 7588, 8729, 9990, 11377, 12896, 14553, 16354, 18305, 20412, 21953, 21960, 21979, 22016, 22077, 22168, 22295, 22464, 22681, 22952, 23283, 23680, 24149, 24696
EXAMPLE
1072 is the sum of two distinct nonzero cubes in exactly one way: 1072 = 7^3 + 9^3. 9 is not in A031980, so 1072 is not the sum of cubes of two distinct earlier terms of A031980 and hence 1072 is in A031980. Therefore 1072 is in not in A141805 and so a term of this sequence. 1729 is the sum of two distinct nonzero cubes in exactly two ways: 1729 = 9^3 + 10^3 = 1^3 + 12^3. 1 and 12 are in A031980, so 1729 is the sum of cubes of two distinct earlier terms of A031980 and hence 1729 is in not A031980. Therefore 1729 is in A141805 and so not a term of this sequence.
CROSSREFS
Cf. A024670, A141805, A031980 (smallest number not occurring earlier and not the sum of cubes of two distinct earlier terms).
a(1) = 1, a(2) = 2, a(n) = smallest number not the sum of 4th powers of 2 distinct earlier terms.
+10
1
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 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, 65, 66, 67, 68, 69, 70, 71, 72, 73
a(1) = 1, a(2) = 2, a(n) = smallest number not the sum of cubes of >= 1 distinct earlier terms.
+10
0
1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 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, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77
REFERENCES
Mihaly Bencze [Beneze], Smarandache recurrence type sequences, Bulletin of pure and applied sciences, Vol. 16E, No. 2, 1997, pp. 231-236.
H. Ibstedt, Smarandache Continued Fractions, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 39-49.
F. Smarandache, Properties of numbers, ASU Special Collections, 1973.
AUTHOR
J. Castillo (arp(AT)cia-g.com) [Broken email address?]
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