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

1,1

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Sequence in context: A046138 A296322 A097279 * A276174 A059455 A373033

Adjacent sequences: A106168 A106169 A106170 * A106172 A106173 A106174

Keyword

nonn

Author

J. M. Bergot, May 19 2007

Extensions

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

Status

approved