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).

1, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 121, 131, 143, 157, 167, 169, 179, 181, 191, 193, 227, 229, 239, 241, 251, 253, 263, 277, 299, 311, 313, 337, 347, 349, 359, 373, 383, 397, 407, 409, 419, 421, 431, 433, 443, 457, 467, 479, 481, 491, 503

Offset: 1

Author

Lekraj Beedassy, Jul 15 2002

Keywords

• nonn

Comments

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

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).

Links

• Frank M Jackson, Table of n, a(n) for n = 1..10000
• R. A. Beauregard and E. R. Suryanarayan, Arithmetic Triangles, Mathematics Magazine, pp. 105-115 70(2) 1997 MAA.

Crossrefs

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

Programs

Maple

isA072330 := proc(n)
    if n = 1 then
        true;
    else
        for p in ifactors(n)[2] do
            if not modp(op(1,p),12) in {1,11} then
                return false ;
            end if;
        end do:
        true;
    end if;
end proc:
for n from 1 to 1000 do
    if isA072330(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 26 2017

Mathematica

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 *)
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
Select[Range[1000], okQ] (* Jean-François Alcover, Mar 06 2020 *)

Formula

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

Extensions

Corrected and extended by Ray Chandler, Jul 02 2004

Incorrect b-file by Carmine Suriano replaced by Frank M Jackson, May 09 2016