A001422 Numbers which are not the sum of distinct squares.

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

Author

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

Keywords

• nonn
• fini
• full

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

• R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.
• T. Sillke, Not the sum of distinct squares
• R. Sprague, Über Zerlegungen in ungleiche Quadratzahlen, Math. Z. 51, (1948), 289-290.
• Eric Weisstein's World of Mathematics, Square Number.
• Index entries for sequences related to sums of squares

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