

A072330


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


8



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
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OFFSET

1,2


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

R. A. Beauregard and E. R. Suryanarayan, Arithmetic Triangles, Mathematics Magazine, pp. 105115 70(2) 1997 MAA.


FORMULA

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


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;


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], 111]&];


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



