%I #11 May 10 2021 07:39:37
%S 24,49,84,184,288,504,628,984,1284,1368,1716,2004,2884,3348,3384,3736,
%T 4368,6484,6816,7288,8004,9508,9808,10200,11508,14584,14836,15684,
%U 19896,21348,21784,22048,25048,25956,27216,27384,35284,38808,40500,40504,44184,47988,49588,50628
%N 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.
%C A subsequence of A254671.
%C Is 49 the only odd term?
%e 24 = 4*4 + 4 + 4.
%e 49 = 9*4 + 9 + 4, and because this is the only representation, 49 is in the sequence.
%e 129 = 4*25+25+4 = 12*9 + 12 + 9, and because 12 is not a square, 129 is not a term.
%o (Python)
%o def isqrt(a):
%o sr = 1 << (int.bit_length(int(a)) >> 1)
%o while a < sr*sr: sr>>=1
%o b = sr>>1
%o while b:
%o s = sr+b
%o if a >= s*s: sr = s
%o b>>=1
%o return sr
%o def isSquare(a):
%o sr = isqrt(a)
%o return a==sr*sr
%o TOP = 100000
%o a = [0]*TOP
%o no= [0]*TOP
%o for y in range(2, TOP//2):
%o for x in range(y, TOP//2):
%o k = x*y + x + y
%o if k>=TOP: break
%o if no[k]==0:
%o a[k]=1
%o if not (isSquare(x) and isSquare(y)):
%o no[k]=1
%o print([n for n in range(TOP) if a[n]>0 and no[n]==0])
%Y Cf. A254671, A256073, A000290.
%K nonn
%O 1,1
%A _Alex Ratushnyak_, May 27 2015
|