The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A283222 Integer area of integer-sided triangle such that the sides are of the form p, p+2, 2(p-1), where p, p+2 and (p-1)/2 are prime numbers. 0
 66, 6810, 182430, 105470250, 17356640970, 678676246650, 1879504308930, 4491035717130, 10618004862030, 21136679055030, 23751520478010, 27081671511090, 27596192489190, 31721097756750, 115248550935750, 133303609919430, 140838829659930, 182797297112430, 197799116497230 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Subsequence of A257049. The area of a triangle (a,b,c) is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2. We observe that the sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2) and Heron's formula gives immediately the area k(2k^2+4) => a(n)= 2*A086381(n)*A253639(n). The corresponding primes p are a subsequence of A056899 (primes of the form n^2+2): 11, 227, 2027, 140627, 4223027, 48650627, 95942027, 171479027, ... We observe that p == 11 mod 72, or p == 11, 83 mod 144. For p>11, p == 27, 227, 627 mod 1000. An interesting property: the greatest prime divisor of a(n) is equal to p. For instance, the prime divisors of 6810 are {2, 3, 5, 227} => p = 227 is the length of the smallest side of the triangle (227, 229, 452). The following table gives the first values of A, the sides of the triangles and the primes (p-1)/2. +-----------+--------+--------+--------+---------+ | A | p | p+2 | 2(p-1)| (p-1)/2 | +-----------+--------+--------+--------+---------+ | 66 | 11 | 13 | 20 | 5 | | 6810 | 227 | 229 | 452 | 113 | | 182430 | 2027 | 2029 | 4052 | 1013 | | 105470250 | 140627 | 140629 | 281252 | 70313 | +-----------+--------+--------+--------+---------+ LINKS Table of n, a(n) for n=1..19. FORMULA a(n) == 6 mod 30. EXAMPLE 66 is in the sequence because the area of the triangle (11, 13, 20) is given by Heron's formula with s = 22 and A = sqrt(22(22-11)(22-13)(22-20)) = 66. The numbers 11, 13 and 5 = (11-1)/2 are primes. MAPLE nn:=100000: for n from 1 by 2 to nn do: if isprime(n^2+2) and isprime(n^2+4) and isprime((n^2+1)/2) then printf(`%d, `, n*(2*n^2+4)): else fi: od: MATHEMATICA nn=10000; lst={}; Do[s=(2*Prime[c]-2+Prime[c+1]+Prime[c])/2; If[IntegerQ[s], area2=s (s-2*Prime[c]+2)(s-Prime[c+1])(s-Prime[c]); If[area2>0&&IntegerQ[Sqrt[area2]] &&Prime[c+1] ==Prime[c]+2 && PrimeQ[(Prime[c]-1)/2], AppendTo[lst, Sqrt[area2]]]], {c, nn}]; Union[lst] PROG (PARI) lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isprime((p-1)/2), ca = p; cb = p+2; cc = 2*(p-1); sp = (ca+cb+cc)/2; a2 = sp*(sp-ca)*(sp-cb)*(sp-cc); if (issquare(a2), print1(sqrtint(a2), ", ")); ); ); } \\ Michel Marcus, Mar 04 2017 CROSSREFS Cf. A056899, A085554, A086381, A188158, A253639, A257049. Sequence in context: A239337 A099639 A003555 * A093266 A282247 A197439 Adjacent sequences: A283219 A283220 A283221 * A283223 A283224 A283225 KEYWORD nonn AUTHOR Michel Lagneau, Mar 03 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 6 20:05 EDT 2023. Contains 363151 sequences. (Running on oeis4.)