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A360790 Squared length of diagonal of right trapezoid with three consecutive prime length sides.

%I A360790 #76 Mar 11 2023 06:22:13
%S A360790 8,13,41,53,137,173,305,397,533,877,977,1373,1697,1885,2245,2813,3517,
%T A360790 3737,4493,5077,5345,6277,6953,7937,9413,10217,10613,11465,12077,
%U A360790 12785,16165,17165,18869,19325,22237,22837,24665,26605,27925,29933,32141,32765,36497,37253,38953,39745
%N A360790 Squared length of diagonal of right trapezoid with three consecutive prime length sides.
%C A360790 The value d is the square of the length of the diagonal of a trapezoid with a height and bases that are consecutive primes, respectively.  The diagonal length is calculated using the Pythagorean theorem, but this distance is squared so that the value is an integer.
%H A360790 Aaron T Cowan, <a href="/A360790/b360790.txt">Table of n, a(n) for n = 1..500</a>
%F A360790 a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2.
%F A360790 a(n) = A001248(n) + A076821(n+1). - _Michel Marcus_, Feb 23 2023
%e A360790         p(2)=3
%e A360790         _ _ _ _
%e A360790 a(1):  |        \  d^2=2^2+(5-3)^2=8
%e A360790 p(1)=2 |_ _ _ _ _\
%e A360790         p(3)=5
%e A360790         p(3)=5
%e A360790         _ _ _ _ _ _
%e A360790 a(2):  |           \    d^2=3^2 + (7-5)^2 = 9+4 = 13
%e A360790 p(2)=3 |            \
%e A360790        |_ _ _ _ _ _ _\
%e A360790         p(4)=7
%e A360790 a(3)= 5^2+(11-7)^2 = 25+16 = 41
%e A360790 a(7)= 17^2+(23-19)^2=305 = 5*61
%t A360790 Map[(#[[1]]^2 + (#[[3]] - #[[2]])^2) &, Partition[Prime[Range[50]], 3, 1]] (* _Amiram Eldar_, Feb 24 2023 *)
%o A360790 (MATLAB) %shorter 1 line version
%o A360790 arrayfun(@(p) p^2+(nextprime(nextprime(p+1)+1)-nextprime(p+1))^2,[primes(10^6)])
%o A360790 (PARI) a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2; \\ _Michel Marcus_, Feb 23 2023
%Y A360790 Cf. A001248, A076821.
%Y A360790 Cf. A131019, A106171.
%K A360790 nonn
%O A360790 1,1
%A A360790 _Aaron T Cowan_, Feb 20 2023