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A085780 Numbers that are a product of 2 triangular numbers. 12
0, 1, 3, 6, 9, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 100, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 190, 198, 210, 216, 225, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 441 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Is there a fast algorithm for detecting these numbers? - Charles R Greathouse IV, Jan 26 2013

The number of rectangles with positive width 1<=w<=i and positive height 1<=h<=j contained in an i*j rectangle is t(i)*t(j), where t(k)=A000217(k), see A096948. - Dimitri Boscainos, Aug 27 2015

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

EXAMPLE

18 = 3*6 = t(2)*t(3) is a product of two triangular numbers and therefore in the sequence.

MAPLE

isA085780 := proc(n)

     local d;

     for d in numtheory[divisors](n) do

        if d^2 > n then

            return false;

        end if;

        if isA000217(d) then

            if isA000217(n/d) then

                return true;

            end if;

        end if;

    end do:

    return false;

end proc:

for n from 1 to 1000 do

    if isA085780(n) then

        printf("%d, ", n) ;

    end if ;

end do: # R. J. Mathar, Nov 29 2015

MATHEMATICA

t1 = Table[n (n+1)/2, {n, 0, 100}]; Select[Union[Flatten[Outer[Times, t1, t1]]], # <= t1[[-1]] &] (* T. D. Noe, Jun 04 2012 *)

PROG

(PARI) A003056(n)=(sqrtint(8*n+1)-1)\2

list(lim)=my(v=List([0]), t); for(a=1, A003056(lim\1), t=a*(a+1)/2; for(b=a, A003056(lim\t), listput(v, t*b*(b+1)/2))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jan 26 2013

CROSSREFS

Cf. A000217, A085782, A068143, A000537 (subsequence), A006011 (subsequence), A033487 (subsequence), A188630 (subsequence).

Sequence in context: A111359 A274428 A085782 * A166047 A223999 A107084

Adjacent sequences:  A085777 A085778 A085779 * A085781 A085782 A085783

KEYWORD

nonn

AUTHOR

Jon Perry, Jul 23 2003

EXTENSIONS

More terms from Max Alekseyev and Jon E. Schoenfield, Sep 04 2009

STATUS

approved

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Last modified September 28 05:32 EDT 2016. Contains 276600 sequences.