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A072330 Common difference n such that primitive triangles exist which are n-arithmetic (i.e., primitive Heronian triangles whose sides in arithmetic progression have common difference n). 8

%I

%S 1,11,13,23,37,47,59,61,71,73,83,97,107,109,121,131,143,157,167,169,

%T 179,181,191,193,227,229,239,241,251,253,263,277,299,311,313,337,347,

%U 349,359,373,383,397,407,409,419,421,431,433,443,457,467,479,481,491,503

%N Common difference n such that primitive triangles exist which are n-arithmetic (i.e., primitive Heronian triangles whose sides in arithmetic progression have common difference n).

%C The first entry in particular is associated with sequences A003500 and A007655.

%C Such a triangle has a middle side 2*x partitioned into x +- 2*n by the corresponding altitude (i.e., median and altitude points are always a distance 2*n apart).

%H Frank M Jackson, <a href="/A072330/b072330.txt">Table of n, a(n) for n = 1..10000</a>

%H R. A. Beauregard and E. R. Suryanarayan, <a href="http://www.jstor.org/stable/2691431">Arithmetic Triangles</a>, Mathematics Magazine, pp. 105-115 70(2) 1997 MAA.

%F n = 1 or a product of primes p congruent to +- 1 (mod 12).

%p isA072330 := proc(n)

%p if n = 1 then

%p true;

%p else

%p for p in ifactors(n)[2] do

%p if not modp(op(1,p),12) in {1,11} then

%p return false ;

%p end if;

%p end do:

%p true;

%p end if;

%p end proc:

%p for n from 1 to 1000 do

%p if isA072330(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Feb 26 2017

%t fac12Q[n_] := And @@ (MemberQ[{1, 11}, #] & /@ Mod[First /@ FactorInteger@ n, 12]); Select[Range[600], fac12Q] (* _Frank M Jackson_, Apr 09 2016 with simplification by _Giovanni Resta_ *)

%t okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];

%t Select[Range[1000], okQ] (* _Jean-Fran├žois Alcover_, Mar 06 2020 *)

%Y Cf. A072360, A086909, A089019, A089020, A096672, A096673, A096674.

%K nonn

%O 1,2

%A _Lekraj Beedassy_, Jul 15 2002

%E Corrected and extended by _Ray Chandler_, Jul 02 2004

%E Incorrect b-file by _Carmine Suriano_ replaced by _Frank M Jackson_, May 09 2016

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Last modified April 5 21:39 EDT 2020. Contains 333260 sequences. (Running on oeis4.)