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

8, 13, 41, 53, 137, 173, 305, 397, 533, 877, 977, 1373, 1697, 1885, 2245, 2813, 3517, 3737, 4493, 5077, 5345, 6277, 6953, 7937, 9413, 10217, 10613, 11465, 12077, 12785, 16165, 17165, 18869, 19325, 22237, 22837, 24665, 26605, 27925, 29933, 32141, 32765, 36497, 37253, 38953, 39745

Offset: 1

Author

Aaron T Cowan, Feb 20 2023

Keywords

• nonn

Comments

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.

Examples

			        p(2)=3
        _ _ _ _
a(1):  |        \  d^2=2^2+(5-3)^2=8
p(1)=2 |_ _ _ _ _\
        p(3)=5
        p(3)=5
        _ _ _ _ _ _
a(2):  |           \    d^2=3^2 + (7-5)^2 = 9+4 = 13
p(2)=3 |            \
       |_ _ _ _ _ _ _\
        p(4)=7
a(3)= 5^2+(11-7)^2 = 25+16 = 41
a(7)= 17^2+(23-19)^2=305 = 5*61
		

Links

• Aaron T Cowan, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A001248, A076821.

Cf. A131019, A106171.

Programs

MATLAB

%shorter 1 line version
arrayfun(@(p) p^2+(nextprime(nextprime(p+1)+1)-nextprime(p+1))^2,[primes(10^6)])

Mathematica

Map[(#[[1]]^2 + (#[[3]] - #[[2]])^2) &, Partition[Prime[Range[50]], 3, 1]] (* Amiram Eldar, Feb 24 2023 *)

PARI

a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2; \\ Michel Marcus, Feb 23 2023

Formula

a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2.

a(n) = A001248(n) + A076821(n+1). - Michel Marcus, Feb 23 2023