A258366 Numbers n representable as x*y + x + y, where x >= y > 1, such that all x's and y's in all representation(s) of n are perfect squares.

24, 49, 84, 184, 288, 504, 628, 984, 1284, 1368, 1716, 2004, 2884, 3348, 3384, 3736, 4368, 6484, 6816, 7288, 8004, 9508, 9808, 10200, 11508, 14584, 14836, 15684, 19896, 21348, 21784, 22048, 25048, 25956, 27216, 27384, 35284, 38808, 40500, 40504, 44184, 47988, 49588, 50628

Offset

1,1

Comments

A subsequence of A254671.

Is 49 the only odd term?

Links

Table of n, a(n) for n=1..44.

Example

24 = 4*4 + 4 + 4.
49 = 9*4 + 9 + 4, and because this is the only representation, 49 is in the sequence.
129 = 4*25+25+4 = 12*9 + 12 + 9, and because 12 is not a square, 129 is not a term.

Prog

(Python)
def isqrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*s:  sr = s
      b>>=1
    return sr
def isSquare(a):
    sr = isqrt(a)
    return a==sr*sr
TOP = 100000
a = [0]*TOP
no= [0]*TOP
for y in range(2, TOP//2):
  for x in range(y, TOP//2):
    k = x*y + x + y
    if k>=TOP: break
    if no[k]==0:
        a[k]=1
        if not (isSquare(x) and isSquare(y)):
            no[k]=1
print([n for n in range(TOP) if a[n]>0 and no[n]==0])

Crossrefs

Cf. A254671, A256073, A000290.

Sequence in context: A044101 A044482 A045294 * A195158 A045279 A042142

Adjacent sequences: A258363 A258364 A258365 * A258367 A258368 A258369

Keyword

nonn

Author

Alex Ratushnyak, May 27 2015

Status

approved