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A085780
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Numbers that are a product of 2 triangular numbers.
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17
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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)
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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EXAMPLE
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18 = 3*6 = t(2)*t(3) is a product of two triangular numbers and therefore in the sequence.
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MAPLE
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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 ;
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MATHEMATICA
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t1 = Table[n (n+1)/2, {n, 0, 100}]; Select[Union[Flatten[Outer[Times, t1, t1]]], # <= t1[[-1]] &] (* T. D. Noe, Jun 04 2012 *)
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PROG
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(PARI) A003056(n)=(sqrtint(8*n+1)-1)\2
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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