A396760 Positive integers k at which A122921(k) - ceiling(sqrt(k/4)) reaches a new record value.

1, 11, 53, 96, 224, 384, 896, 1536, 2816, 3584, 6144, 11264, 14336, 24576, 45056, 57344, 98304, 180224, 229376, 393216, 720896, 917504, 1572864, 2883584, 3670016, 6291456, 11534336, 14680064, 25165824, 46137344, 58720256, 100663296, 184549376, 234881024, 402653184, 738197504, 939524096

Offset

1,2

Comments

A122921(k) is the least possible largest part x in a representation k = x^2+y^2+z^2+w^2 with x>=y>=z>=w>=0. Since x >= ceiling(sqrt(k/4)) always, the gap is nonnegative and measures how far k is forced from four equal squares; this sequence lists the positive k where the gap reaches a new maximum.

This sequence partially overlaps A006431, the numbers with a unique partition into four nonnegative squares, but is distinct: 53 and 2816 are terms here and are not in A006431, while 512 is in A006431 and is not a term here.

A285552 is related but distinct: it concerns the largest possible part measured from floor(sqrt(k)); here the largest part is minimized and the gap is measured from ceiling(sqrt(k/4)).

Links

Table of n, a(n) for n=1..37.

Nenad Micic, Author's repository (retained scan, decompositions, b-file, programs)

Formula

Conjecture: the sequence consists exactly of 1, 11, 53 and the three chains 96*4^m, 224*4^m, 2816*4^m (m >= 0), merged in increasing order. This formula gives all 37 listed terms below 10^9; beyond that bound it is conjectural.

Example

For k = 1, 1 = 1^2+0^2+0^2+0^2, so A122921(1) = 1, ceiling(sqrt(1/4)) = 1, and the gap is 0.
For k = 11, 11 = 3^2+1^2+1^2+0^2, so A122921(11) = 3, ceiling(sqrt(11/4)) = 2, and the gap is 1, a new record.
For k = 53, 53 = 6^2+3^2+2^2+2^2, so A122921(53) = 6, ceiling(sqrt(53/4)) = 4, and the gap is 2, a new record.

Prog

(PARI)
  ceilq(k) = {
    my(x=sqrtint((k+3)\4));
    while(4*x^2 < k, x++);
    x
  };
  amin(k) = {
    my(xs=sqrtint(k));
    for(x=ceilq(k), xs,
      my(r1=k-x^2);
      for(y=0, min(x, sqrtint(r1)),
        my(r2=r1-y^2);
        for(z=0, min(y, sqrtint(r2)),
          my(r3=r2-z^2);
          my(w=sqrtint(r3));
          if(w<=z && w*w==r3, return(x))
        )
      )
    )
  };
  g(k) = amin(k) - ceilq(k);
  lista(N) = {
    my(rec=-1);
    for(k=1, N,
      my(gk=g(k));
      if(gk>rec, rec=gk; print1(k, ", "))
    )
  };

Crossrefs

Cf. A122921, A122922, A122923, A122924, A006431, A285552, A178786.

Sequence in context: A004622 A045471 A086715 * A178453 A225582 A164580

Adjacent sequences: A396755 A396756 A396758 * A396765 A396768 A396770

Keyword

nonn,new

Author

Nenad Micic, Jun 04 2026

Status

approved