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The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10 , the digital root of a nonzero triangular number is always 1, 3, 6, or 9.
All square triangular numbers have the form , where is a convergent to the continued fraction expansion of , the square root of 2. [4]A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the th triangular number (+) is square, then so is the larger (+) th triangular number, since:
The sequence of highly composite numbers (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS). Highly composite numbers whose number of divisors is also a highly composite number are. 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600 ...
The sequence of squared triangular numbers is [1] 0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, ... . These numbers can be viewed as figurate numbers, a four-dimensional hyperpyramidal generalization of the triangular numbers and square pyramidal numbers. As Stein (1971) observes, these numbers also count the number of ...
Ternary: The base-three numeral system with 0, 1, and 2 as digits. Quaternary: The base-four numeral system with 0, 1, 2, and 3 as digits. Hexadecimal: Base 16, widely used by computer system designers and programmers, as it provides a more human-friendly representation of binary-coded values.
Square number. Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.
All centered square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in base 6, 8, and 12. Every centered square number except 1 is the hypotenuse of a Pythagorean triple (3-4-5, 5-12-13, 7-24-25, ...). This is exactly the sequence of Pythagorean ...
e. Quaternary / kwəˈtɜːrnəri / is a numeral system with four as its base. It uses the digits 0, 1, 2, and 3 to represent any real number. Conversion from binary is straightforward. Four is the largest number within the subitizing range and one of two numbers that is both a square and a highly composite number (the other being thirty-six ...