cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A047700 Numbers that are the sum of 5 positive squares.

Original entry on oeis.org

5, 8, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 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, 74, 75, 76, 77, 78, 79
Offset: 1

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Author

Arlin Anderson (starship1(AT)gmail.com)

Keywords

Comments

Complement of A047701.

Examples

			From _David A. Corneth_, Aug 04 2020: (Start)
2009 is in the sequence as 2009 = 18^2 + 18^2 + 18^2 + 19^2 + 26^2.
2335 is in the sequence as 2335 = 19^2 + 19^2 + 20^2 + 22^2 + 27^2.
3908 is in the sequence as 3908 = 24^2 + 24^2 + 26^2 + 28^2 + 36^2. (End)

Links

Crossrefs

Cf. A000414, A003328, A047701, A294675, A344795, A344805.

Formula

a(n) = n + 12 for n >= 22. - David A. Corneth, Aug 04 2020

A003334 Numbers that are the sum of 11 positive cubes.

Original entry on oeis.org

11, 18, 25, 32, 37, 39, 44, 46, 51, 53, 58, 60, 63, 65, 67, 70, 72, 74, 77, 79, 81, 84, 86, 88, 89, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109, 110, 112, 114, 115, 116, 117, 119, 121, 122, 123, 124, 126, 128, 129, 130, 131, 133, 135, 136, 137, 138, 140, 141, 142, 143, 144
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
The sequence contains all integers greater than 321 which is the last of only 92 positive integers not in this sequence. - M. F. Hasler, Aug 25 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
1120 is in the sequence as 1120 = 2^3 + 3^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 5^3 +  8^3.
2339 is in the sequence as 2339 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 9^3 +  9^3.
3594 is in the sequence as 3594 = 4^3 + 5^3 + 6^3 + 6^3 + 6^3 + 6^3 + 7^3 + 7^3 + 7^3 + 8^3 + 10^3. (End)

Links

Crossrefs

Other sequences S(k, m) of numbers that are the sum of k nonzero m-th powers:
A000404 = S(2, 2), A000408 = S(3, 2), A000414 = S(4, 2) complement of A000534,
A047700 = S(5, 2) complement of A047701, A180968 = complement of S(6,2);
A003325 = S(2, 3), A003072 = S(3, 3), A003327 .. A003335 = S(4 .. 12, 3) and A332107 .. A332111 = complement of S(7 .. 11, 3);
A003336 .. A003346 = S(2 .. 12, 4), A003347 .. A003357 = S(2 .. 12, 5),
A003358 .. A003368 = S(2 .. 12, 6), A003369 .. A003379 = S(2 .. 12, 7),
A003380 .. A003390 = S(2 .. 12, 8), A003391 .. A004801 = S(2 .. 12, 9),
A004802 .. A004812 = S(2 .. 12, 10), A004813 .. A004823 = S(2 .. 12, 11).

Programs

  • PARI
    (A003334_upto(N, k=11, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ See also A003333 for alternate code. - M. F. Hasler, Aug 03 2020

Formula

a(n) = n + 92 for all n > 229. - M. F. Hasler, Aug 25 2020

A123120 Numbers k>0 such that m+k is not the sum of m nonzero squares for any m>5.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 13
Offset: 1

Views

Author

Alexander Adamchuk, Sep 28 2006

Keywords

Comments

Also, numbers which are not the sum of "squares-minus-1" (cf. A005563). - Benoit Jubin, Apr 14 2010
Conjecture: All but (n+6) positive numbers are equal to the sum of n>5 nonzero squares. For all n>5 the only (n+6) positive numbers that are not equal to the sum of n nonzero squares are {1,2,3,...,n-3,n-2,n-1,n+1,n+2,n+4,n+5,n+7,n+10,n+13}.
Numbers that are not squares (n=1): A000037 = {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28,...}
Numbers that are not the sum of 2 nonzero squares (n=2): A018825 = {1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 16, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 36,...}.
Numbers that are not the sum of three nonzero squares (n=3): A004214 = {1, 2, 4, 5, 7, 8, 10, 13, 15, 16, 20, 23, 25, 28, 31, 32, 37, 39, 40, 47, 52, 55, 58,...}.
Numbers that are not the sum of 4 nonzero squares (n=4): A000534 ={1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512,...}.
Numbers that are not the sum of 5 nonzero squares (n=5): A047701 = {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33}.
Numbers that are not the sum of 6 nonzero squares (n=6): {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 19}.
Numbers that are not the sum of 7 nonzero squares (n=7): {1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20}.
Numbers that are not the sum of 8 nonzero squares (n=8): {1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 18, 21}.
Numbers that are not the sum of 9 nonzero squares (n=9): {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 19, 22}.
The above conjecture appears as Theorem 2 on p. 73 in the Grosswald reference, where it is attributed to E. Dubouis (1911). - Wolfdieter Lang, Mar 27 2013

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, pp. 73-74.

Crossrefs

Cf. A000037, A018825, A004214, A000534, A047701.

Extensions

Edited by M. F. Hasler, Feb 23 2018

A180968 The only integers that cannot be partitioned into a sum of six positive squares.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 19
Offset: 1

Views

Author

Ant King, Sep 30 2010

Keywords

Comments

From R. J. Mathar, Sep 11 2012: (Start)
Not the sum of 7 positive squares: 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20.
Not the sum of 8 positive squares: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 18, 21.
Not the sum of 9 positive squares: 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 19, 22.
Not the sum of 10 positive squares: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 20, 23. (End)

Examples

			As the sixth integer which cannot be partitioned into a sum of six positive squares is 7, we have a(6)=7.

References

  • Dubouis, E.; L'Interm. des math., vol. 18, (1911), pp. 55-56, 224-225.
  • Grosswald, E.; Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), pp.73-74.

Links

  • Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18.

Crossrefs

Cf. A047701 (not the sum of 5 squares)

Programs

  • Mathematica
    s=6;B={1,2,4,5,7,10,13}; Union[Range[s-1],s+B]//Sort

Formula

Let B be the set of integers {1,2,4,5,7,10,13}. Then, for s>=6, every integer can be partitioned into a sum of s positive squares except for 1,2,...,s-1 and s+b where b is a member of the set B [Dubouis].

A223729 Numbers appearing in a theorem on the representation of numbers as sums of five non-vanishing squares.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 13, 28
Offset: 1

Views

Author

Wolfdieter Lang, Mar 27 2013

Keywords

Comments

See A047701 for the positive numbers that are not the sum of five nonzero squares, which are precisely 1, 2, 3, 4 and 5 + a(n), n = 1,...,8. This appears as Theorem 2. in Grosswald's book on p. 73-4 with references to E. Dubouis (1911) and some others.

References

  • E. Dubouis, Solution of a problem of J. Tannery, L'Intermédiaire Math. 18 (1911) 55-56.
  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 73-74.

Crossrefs

Cf. A123120, A047701.

A295796 The only integers that cannot be partitioned into a sum of seven positive squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20
Offset: 1

Views

Author

Robert Price, Nov 27 2017

Keywords

References

  • Dubouis, E.; L'Interm. des math., vol. 18, (1911), pp. 55-56, 224-225.
  • Grosswald, E.; Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), pp.73-74.

Links

  • Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18.

Crossrefs

Cf. A025431, A047701, A180968.
Showing 1-6 of 6 results.

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