A106171 A triangle with three consecutive primes as sides has an area that is a prime after rounding. The sequence gives the first of the three consecutive primes.

5, 11, 23, 59, 71, 89, 211, 239, 269, 349, 389, 419, 431, 467, 479, 521, 571, 577, 647, 863, 983, 1087, 1213, 1223, 1733, 1747, 1759, 1933, 1949, 1973, 2131, 2297, 2411, 2521, 2659, 2879, 2909, 2999, 3011, 3191, 3203, 3209, 3391, 3467, 3469, 3517, 3559

Offset: 1

Author

J. M. Bergot, May 19 2007

Keywords

• nonn

Examples

			For sides 5,7,11 the formula gives 12.96 and with rounding this becomes 13, a prime.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Programs

Maple

s:=proc(n) local a,b,c,p,A: a:=ithprime(n): b:=ithprime(n+1): c:=ithprime(n+2): p:=(a+b+c)/2: A:=sqrt(p*(p-a)*(p-b)*(p-c)): if isprime(round(A))=true then a else fi end: seq(s(n),n=1..700); # Emeric Deutsch, May 25 2007
Digits := 60 : isA106171 := proc(p) local q,r,s,area ; if isprime(p) then q := nextprime(p) ; r := nextprime(q) ; s := (p+q+r)/2 ; area := round(sqrt(s*(s-p)*(s-q)*(s-r))) ; RETURN(isprime(area)) ; else false ; fi ; end: for n from 1 to 900 do p := ithprime(n) : if isA106171(p) then printf("%d,",p) ; fi ; od : # R. J. Mathar, Jun 08 2007

Mathematica

arQ[{a_,b_,c_}]:=With[{s=(a+b+c)/2},PrimeQ[Round[Sqrt[s(s-a)(s-b)(s-c)]]]]; Select[Partition[Prime[Range[600]],3,1],arQ][[;;,1]] (* Harvey P. Dale, Jul 13 2025 *)

Formula

Simply use the formula for the area of a triangle given the three sides.

Extensions

More terms from Emeric Deutsch and R. J. Mathar, May 25 2007