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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
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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;

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 *)

CROSSREFS

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

Sequence in context: A048393 A136058 A106073 * A097933 A166484 A127043

Adjacent sequences:  A072327 A072328 A072329 * A072331 A072332 A072333

KEYWORD

nonn

AUTHOR

Lekraj Beedassy, Jul 15 2002

EXTENSIONS

Corrected and extended by Ray Chandler, Jul 02 2004

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

STATUS

approved

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Last modified October 21 13:24 EDT 2019. Contains 328299 sequences. (Running on oeis4.)