A031980 -id:A031980 - cp's OEIS frontend

A141805 Complement of A031980.

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

Offset: 1

Klaus Brockhaus, Jul 16 2008

nonn

Subsequence of A024670; A141806 gives the terms of A024670 that are not in this sequence.

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.

			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 is 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.

Klaus Brockhaus, Table of n, a(n) for n = 1..24834
Index to sequences related to sums of squares and sums of cubes

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).

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;

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 *)

A292740 Indices k such that A292547(k) = 0.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Sep 22 2017

nonn

Complement of A290276.

Conjecture: for k > 212594 there are no more terms in this sequence (tested for k < 63000000).

			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

Vaclav Kotesovec, Table of n, a(n) for n = 1..54766

Cf. A276517, A279329, A279486, A279529, A290276, A292547.

Cf. A001476, A022555, A031980.

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 *)

A141806 Terms of A024670 that are not in A141805.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jul 16 2008

nonn

A024670 gives the sums of cubes of two distinct positive integers. A141805, the complement of A031980, is a subsequence of A024670.

			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.

Klaus Brockhaus, Table of n, a(n) for n = 1..2774
Index to sequences related to sums of squares and sums of cubes

Cf. A024670, A141805, A031980 (smallest number not occurring earlier and not the sum of cubes of two distinct earlier terms).

A133137 a(1) = 1, a(2) = 2, a(n) = smallest number not the sum of 4th powers of 2 distinct earlier terms.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Sep 21 2007

This is to A000583 as A031980 is to A000578. First differs from complement of A088687 at 83522 = 1^4 + 17^4.

Cf. A000583, A031980, A088687.

A031981 a(1) = 1, a(2) = 2, a(n) = smallest number not the sum of cubes of >= 1 distinct earlier terms.

Original entry on oeis.org

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

Offset: 1

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

nonn

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.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences

Essentially the same as A001476. See also A031980.

More terms from Sean A. Irvine, May 04 2020

cp's OEIS Frontend

A141805 Complement of A031980.

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A292740 Indices k such that A292547(k) = 0.

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A141806 Terms of A024670 that are not in A141805.

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A133137 a(1) = 1, a(2) = 2, a(n) = smallest number not the sum of 4th powers of 2 distinct earlier terms.

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A031981 a(1) = 1, a(2) = 2, a(n) = smallest number not the sum of cubes of >= 1 distinct earlier terms.

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