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A274697 Variation on Fermat's Diophantine m-tuple: 1 + the GCD of any two distinct terms is a square.

%I A274697 #10 Mar 14 2020 11:25:11
%S A274697 0,3,15,24,48,63,120,195,255,528,960,3024,3363,3480,3720,3843,4095,
%T A274697 4623,5475,12099,16383,19599,24963,37635,38415,44943,56643,62499,
%U A274697 65535,69168,71823,85263,94863,114243,168099
%N A274697 Variation on Fermat's Diophantine m-tuple: 1 + the GCD of any two distinct terms is a square.
%C A274697 a(1) = 0; for n>1, a(n) = smallest integer > a(n-1) such that GCD(a(n),a(i))+1 is square for all 1 <= i <= n-1.
%e A274697 After a(1)=0, a(2)=3, a(3)=15, we want m, the smallest number > 15 such that GCD(0,m)+1, GCD(3,m)+1 and GCD(15,m)+1 are squares: this is m = 24 = a(4).
%o A274697 (Sage)
%o A274697 seq = []
%o A274697 prev_element = 0
%o A274697 seq.append( prev_element )
%o A274697 max_n = 35
%o A274697 for n in range(2, max_n+1):
%o A274697     next_element = prev_element + 1
%o A274697     while True:
%o A274697         all_match = True
%o A274697         for element in seq:
%o A274697             x = gcd( element, next_element ) + 1
%o A274697             if not ( is_square(x) ):
%o A274697                 all_match = False
%o A274697                 break
%o A274697         if all_match:
%o A274697             seq.append( next_element )
%o A274697             print(seq)
%o A274697             break
%o A274697         next_element = next_element + 1
%o A274697     prev_element = next_element
%o A274697 print(seq)
%Y A274697 Cf. A030063.
%K A274697 nonn
%O A274697 1,2
%A A274697 _Robert C. Lyons_, Jul 05 2016