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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).

%I A072330 #38 Mar 06 2020 08:08:57
%S A072330 1,11,13,23,37,47,59,61,71,73,83,97,107,109,121,131,143,157,167,169,
%T A072330 179,181,191,193,227,229,239,241,251,253,263,277,299,311,313,337,347,
%U A072330 349,359,373,383,397,407,409,419,421,431,433,443,457,467,479,481,491,503
%N 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).
%C A072330 The first entry in particular is associated with sequences A003500 and A007655.
%C A072330 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 A072330 Frank M Jackson, <a href="/A072330/b072330.txt">Table of n, a(n) for n = 1..10000</a>
%H A072330 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 A072330 n = 1 or a product of primes p congruent to +- 1 (mod 12).
%p A072330 isA072330 := proc(n)
%p A072330     if n = 1 then
%p A072330         true;
%p A072330     else
%p A072330         for p in ifactors(n)[2] do
%p A072330             if not modp(op(1,p),12) in {1,11} then
%p A072330                 return false ;
%p A072330             end if;
%p A072330         end do:
%p A072330         true;
%p A072330     end if;
%p A072330 end proc:
%p A072330 for n from 1 to 1000 do
%p A072330     if isA072330(n) then
%p A072330         printf("%d,",n) ;
%p A072330     end if;
%p A072330 end do: # _R. J. Mathar_, Feb 26 2017
%t A072330 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 A072330 okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
%t A072330 Select[Range[1000], okQ] (* _Jean-François Alcover_, Mar 06 2020 *)
%Y A072330 Cf. A072360, A086909, A089019, A089020, A096672, A096673, A096674.
%K A072330 nonn
%O A072330 1,2
%A A072330 _Lekraj Beedassy_, Jul 15 2002
%E A072330 Corrected and extended by _Ray Chandler_, Jul 02 2004
%E A072330 Incorrect b-file by _Carmine Suriano_ replaced by _Frank M Jackson_, May 09 2016