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A241855 Array t(n,k) of sum of successive even powers of primes, where t(n,k) = sum_(j=0..k-1) prime(n)^(2j), with n>=1 and k>=0, read by ascending antidiagonals.

%I A241855 #18 May 05 2014 09:17:21
%S A241855 0,0,1,0,1,5,0,1,10,21,0,1,26,91,85,0,1,50,651,820,341,0,1,122,2451,
%T A241855 16276,7381,1365,0,1,170,14763,120100,406901,66430,5461,0,1,290,28731,
%U A241855 1786324,5884901,10172526,597871,21845,0,1,362,83811,4855540,216145205,288360150,254313151,5380840,87381
%N A241855 Array t(n,k) of sum of successive even powers of primes, where t(n,k) = sum_(j=0..k-1) prime(n)^(2j), with n>=1 and k>=0, read by ascending antidiagonals.
%C A241855 Conjecture: any term, except 0 and 1, is never a square.
%C A241855 Row n=1 is A002450,
%C A241855 row n=2 is A002452,
%C A241855 row n=3 is A218728,
%C A241855 row n=4 is A218753,
%C A241855 rows n>=5 are not in the OEIS,
%C A241855 column k=2 is A066872,
%C A241855 columns k>=3 are not in the OEIS.
%F A241855 t(n,k) = ((prime(n)^2)^k-1)/(prime(n)^2-1).
%e A241855 Array begins:
%e A241855 0,  1,   5,    21,      85,       341,        1365, ...
%e A241855 0,  1,  10,    91,     820,      7381,       66430, ...
%e A241855 0,  1,  26,   651,   16276,    406901,    10172526, ...
%e A241855 0,  1,  50,  2451,  120100,   5884901,   288360150, ...
%e A241855 0,  1, 122, 14763, 1786324, 216145205, 26153569806, ...
%e A241855 etc.
%t A241855 t[n_, k_] := ((Prime[n]^2)^k-1)/(Prime[n]^2-1); Table[t[n-k+1, k], {n, 0, 10}, {k, 0, n}] // Flatten
%Y A241855 Cf. A002450, A002452, A066872, A059826, A059830, A059839, A218728, A218753.
%K A241855 nonn,tabl
%O A241855 1,6
%A A241855 _Jean-François Alcover_, Apr 30 2014