Trapezoidal numbers are whole numbers that can be written as the sum of two or more consecutive positive integers. For example, 9 is a trapezoidal number because it can be written as 2+3+4. In fact, 9 is doubly trapezoidal because it can also be written as 4+5. (By the way, triangular numbers are also trapezoidal. I’m a fan of inclusive definitions like these.)
What is the smallest number that is triply trapezoidal, meaning it is trapezoidal in three different ways?
Extra credit: What is the smallest number that is quadruply trapezoidal? Quintuply trapezoidal? Sextuply trapezoidal?
The sum of an even number of consecutive positive integers will be:
The sum of an odd number of consecutive positive integers will be:
$a + (a+1) = 2a+1$. It follows every odd number $\ge 3$ is trapezoidal.
In order to find the smallest of each type, only odd numbers of consecutive positive integers will be examined.
$b + (b+1) + (b+2) = 3b+3$. It follows every odd number divisible by $\ge 9$ is doubly trapezoidal.
Note that $9$ is the smallest odd prime squared.
$c + (c+1) + (c+2) + (c+3) + (c+4) = 5c+10$.
Finding the smallest $b,c$ for
$$3b + 3 = 5c + 10$$
$$3(b+1) = 5(c+2)$$
yields $b = 4, c= 1$.
$15$ is the smallest triply trapezoidal number.
$$7+8 = 15$$$$4+5+6 = 15$$$$1+2+3+4+5 = 15$$Note that $15 = p_1 \cdot p_2$, that is, the smallest product of two distinct odd primes.
Using $p_1^2 \cdot p_2$, derived from above, I examined 45.
$$22+23 = 45$$$$14+15+16 = 45$$$$7+8+9+10+11 = 45$$$$5+6+7+8+9+10 = 45$$$$1+2+3+4+5+6+7+8+9 = 45$$$45$ is the smallest quintuply trapezoidal number.
Using $p_1^3 \cdot p_2$, derived from above, I examined 135.
$$44+45+46 = 135$$$$25+26+27+28+29 = 135$$$$20+21+22+23+24+25 = 135$$$$11+12+13+14+15+16+17+18+19 = 135$$$$9+10+11+12+13+14+15+16+17+18 = 135$$$$2+3+4+5+6+7+8+9+10+11+12+13+14+15+16 = 135$$$135$ is the smallest sextuply trapezoidal number.
Extending to $p_1 \cdot p_2 \cdot p_3$, derived from above, I examined 105.
$$52+53 = 105$$$$34+35+36 = 105$$$$19+20+21+22+23 = 105$$$$15+16+17+18+19+20 = 105$$$$12+13+14+15+16+17+18 = 105$$$$6+7+8+9+10+11+12+13+14+15 = 105$$$$1+2+3+4+5+6+7+8+9+10+11+12+13+14 = 105$$$105$ is the smallest septuply trapezoidal number.