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A274694 Variation on Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a prime power.

%I A274694 #11 Mar 14 2020 17:15:42
%S A274694 1,2,3,4,6,12,211050,3848880,20333040,125038830,2978699430
%N A274694 Variation on Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a prime power.
%C A274694 a(1) = 1; for n>1, a(n) = smallest integer > a(n-1) such that a(n)*a(i)+1 is a prime power for all 1 <= i <= n-1.
%e A274694 After a(1)=1, a(2)=2, a(3)=3, we want m, the smallest number > 3 such that m+1, 2m+1 and 3m+1 are all prime powers: this is m = 4 = a(4).
%o A274694 (Sage)
%o A274694 seq = [1]
%o A274694 prev_element = 1
%o A274694 max_n = 8
%o A274694 for n in range(2, max_n+1):
%o A274694     next_element = prev_element + 1
%o A274694     while True:
%o A274694         all_match = True
%o A274694         for element in seq:
%o A274694             x = element * next_element + 1
%o A274694             if not x.is_prime_power():
%o A274694                 all_match = False
%o A274694                 break
%o A274694         if all_match:
%o A274694             seq.append( next_element )
%o A274694             break
%o A274694         next_element += 1
%o A274694     prev_element = next_element
%o A274694 print(seq)
%Y A274694 Cf. A030063, A034881, A246655.
%K A274694 nonn,more
%O A274694 1,2
%A A274694 _Robert C. Lyons_, Jul 02 2016