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 1-4 of 4 results.

A001422 Numbers which are not the sum of distinct squares.

Original entry on oeis.org

2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128
Offset: 1

Views

Author

N. J. A. Sloane, Jeff Adams (jeff.adams(AT)byu.net)

Keywords

Comments

This is the complete list (Sprague).

References

  • S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
  • Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
  • J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 222.

Links

Crossrefs

Complement of A003995. Subsequence of A004441.
Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A121405 (corresponding sequences for triangular and pentagonal numbers)
Cf. A033461, A276517.
Cf. A001476, A046039, A194768, A194769 for 3rd, 4th, 5th, 6th powers.
Cf. A001661.

Programs

  • Mathematica
    nn=50; t=Rest[CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)
  • PARI
    select( is_A001422(n,m=n)={m^2>n&& m=sqrtint(n); n!=m^2&&!while(m>1,isSumOfSquares(n-m^2,m--)&&return)}, [1..128]) \\ M. F. Hasler, Apr 21 2020

A004195 The numbers not expressible as the sum of 4 distinct nonzero squares can be written D*4^n union E. This is D.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 22, 23, 25, 26, 27, 31, 33, 34, 37, 38, 42, 43, 47, 55, 58, 67, 73, 82, 97, 103
Offset: 1

Views

Author

N. J. A. Sloane, Dan Hoey

Keywords

Links

Crossrefs

Cf. A004196, A004441.

A004196 The numbers not expressible as the sum of 4 distinct nonzero squares can be written D*4^n union E. This is E.

Original entry on oeis.org

21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157
Offset: 1

Views

Author

N. J. A. Sloane, Dan Hoey

Keywords

Links

Crossrefs

Cf. A004195, A004441.

A175958 Number of partitions of n^2 into 4 distinct nonzero squares.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 5, 0, 4, 4, 5, 0, 10, 4, 7, 0, 11, 7, 17, 1, 13, 17, 15, 0, 29, 13, 27, 4, 23, 17, 41, 0, 29, 35, 32, 4, 66, 24, 38, 0, 47, 35, 73, 7, 50, 56, 73, 1, 91, 42, 63, 17, 68, 49, 125, 0, 103, 93, 83, 13, 133, 86, 93, 4
Offset: 0

Views

Author

R. J. Mathar, Oct 30 2010

Keywords

Examples

			a(9)=1 refers to the partition 9^2 = 2^2+4^2+5^2+6^2. a(11)=1 refers to 11^2 = 1^2+2^2+4^2+10^2. a(13)=2 refers to 13^2 = 1^2+2^2+8^2+10^2 = 2^2+4^2+7^2+10^2.

Links

Crossrefs

Cf. A004195, A004441, A004433, A181524.

Programs

  • Maple
    A025443 := proc(n) local res,a,b,c,d ; res := 0 ; for a from 1 do if 4*a^2 > n then break; fi; for b from a+1 do if a^2+3*b^2 > n then break; fi; for c from b+1 do if a^2+b^2+2*c^2 > n then break; fi; for d from c+1 do if a^2+b^2+c^2+d^2 > n then break; elif a^2+b^2+c^2+d^2 = n then res := res+1 ; fi ; end do; end do; end do: end do: res ; end proc:
A := proc(n) A025443(n^2) ; end proc: seq(A(n),n=0..60) ;
# second Maple program:
b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
      `if`(t*i^2n, 0, b(n-i^2,i-1,t-1))))
    end:
a:= n-> b(n^2, n, 4):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 07 2013
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[t*i^2 < n, 0, If[i == 1, 0, b[n, i-1, t]] + If[i^2 > n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n^2, n, 4]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 24 2015, after Alois P. Heinz *)

Formula

a(n) = A025443(n^2).

Extensions

More terms from Alois P. Heinz, Feb 07 2013
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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