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 A085780 Numbers that are a product of 2 triangular numbers. 17
 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 (Python) from itertools import count, islice from sympy import divisors, integer_nthroot def A085780_gen(startvalue=0): # generator of terms if startvalue <= 0: yield 0 for n in count(max(startvalue, 1)): for d in divisors(m:=n<<2): if d**2 > m: break if integer_nthroot((d<<2)+1, 2)[1] and integer_nthroot((m//d<<2)+1, 2)[1]: yield n break A085780_list = list(islice(A085780_gen(), 10)) # Chai Wah Wu, Aug 28 2022 CROSSREFS Cf. A000217, A085782, A068143, A000537 (subsequence), A006011 (subsequence), A033487 (subsequence), A188630 (subsequence). Cf. A072389 (this times 4). Sequence in context: A274428 A344158 A085782 * A166047 A310141 A348550 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 December 2 08:12 EST 2023. Contains 367515 sequences. (Running on oeis4.)