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Square of a triangular number
In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, 1 3 + 2 3 + 3 3 + ⋯ + n 3 = ( 1 + 2 + 3 + ⋯ + n ) 2
Squared_triangular_number
Integer that is both a perfect square and a triangular number
mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words
Square_triangular_number
Figurate number
arranged in an equilateral triangle. The triangular lattice representing the n {\displaystyle n} th triangular number contains n {\displaystyle n} rows: the
Triangular_number
Product of an integer with itself
squared". The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (1 × 1). Hence, a square with
Square_number
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns
Pentagonal_number
Natural number
non-trivial square triangular number. Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than
36_(number)
Number used to approximate the square root of 2
As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations
Pell_number
Type of figurate number
properties of oblong, triangular, and square numbers. The number 10 for example, can be arranged as a triangle (see triangular number): But 10 cannot be
Polygonal_number
Natural number
152), an octagonal number, and a squared triangular number (225 = (1 + 2 + 3 + 4 + 5)2 = 13 + 23 + 33 + 43 + 53) . As the square of a double factorial
225_(number)
Figure that, added to a given figure, makes a larger figure of the same shape
multiplication table proves the Nicomachus theorem, claiming that each squared triangular number is a sum of consecutive cubes. In an acute isosceles triangle
Gnomon_(figure)
Figurate number
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers
Pyramidal_number
Natural number
432,881 = 403912, square triangular number 1,673,196,525 : Least common multiple of the odd integers from 1 to 25 1,677,922,740 : number of series-reduced
1,000,000,000
Indian mathematician-astronomer (476–550)
{\displaystyle 1^{3}+2^{3}+\cdots +n^{3}=(1+2+\cdots +n)^{2}} (see squared triangular number) Aryabhata's system of astronomy was called the audAyaka system
Aryabhata
Natural number
7-digit prime number 1,000,405 = Smallest triangular number with 7 digits and the 1,414th triangular number 1,002,001 = 10012, palindromic square 1,006,003
1,000,000
Centered figurate number that represents a triangle with a dot in the center
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other
Centered_triangular_number
Sum of a factorial number and a triangular number
In number theory, a factoriangular number is an integer formed by adding a factorial and a triangular number with the same index. The name is a portmanteau
Factoriangular_number
Number of stacked spheres in a pyramid
n-th square pyramidal number. The number of rectangles in a square grid is given by the squared triangular numbers. The square pyramidal number P n {\displaystyle
Square_pyramidal_number
Type of triangular number
n} th triangular number, then the doubly triangular numbers are the numbers of the form T T n {\displaystyle T_{T_{n}}} . The doubly triangular numbers
Doubly_triangular_number
Number, product of consecutive integers
triangular number and n more than the nth square number, as given by the alternative formula n2 + n for pronic numbers. Hence the nth pronic number and
Pronic_number
Class of series of figurate numbers, each having a central dot
initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the
Centered_polygonal_number
Prism with a 3-sided base
base's edges equals the number of its square faces. More generally, the triangular prism is uniform. This means that a triangular prism has regular faces
Triangular_prism
Natural number
is the fourth square triangular number. As a figurate number, 204 is also a nonagonal number and a truncated triangular pyramid number. 204 is a member
204_(number)
Natural number
prime number 10,001,628 = Smallest triangular number with 8 digits and the 4,472nd triangular number 10,004,569 = 31632, the smallest 8-digit square 10,077
10,000,000
Natural number
pyramidal number and a dodecagonal number. Additionally, it is the index, in the sequence of triangular numbers, of the fifth square triangular number: 41616
288_(number)
Polyhedral number representing a tetrahedron
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron
Tetrahedral_number
Size of a geometric arrangement of points
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes
Figurate_number
Number of dots in a centred dot square
In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center
Centered_square_number
Natural number
of 100 and also the square of 100,000. 10,000,000,019 = smallest 11-digit prime number. 10,000,020,331 = smallest triangular number with 11 digits and
10,000,000,000
Mathematical problem of square numbers which are also square-pyramidal
are both tetrahedral and square pyramidal. Square triangular number, the numbers that are simultaneously square and triangular Close-packing of equal spheres
Cannonball_problem
Number equal to the sum of its proper divisors
2^{p-1}(2^{p}-1)} , each even perfect number is the ( 2 p − 1 ) {\displaystyle (2^{p}-1)} -th triangular number (and hence equal to the sum of the integers
Perfect_number
Centered figurate number
numbers. Geometrically, the nth star number is made up of a central point and 12 copies of the (n−1)th triangular number — making it numerically equal to
Star_number
Polyhedron with eight triangular faces
Augmented triangular prism: The result of gluing a triangular prism to a square pyramid, this has six equilateral triangle faces and two square faces. It
Octahedron
Natural number
second and only prime triangular number, and Carl Friedrich Gauss proved that every integer is the sum of at most three triangular numbers. Three is the
3
Number divisible only by 1 and itself
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that
Prime_number
Natural number
J. A. (ed.). "Sequence A000537 (Sum of first n cubes; or n-th triangular number squared.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
9
Iterative algorithm on numbers
In number theory, Kaprekar’s routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with
Kaprekar's_routine
Number that remains the same when its digits are reversed
A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are
Palindromic_number
Class of binary number
In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of
Evil_number
Natural number
constant 6181 – octahedral number 6200 – harmonic divisor number 6201 – square pyramidal number 6216 – triangular number 6217 – super-prime, prime of
6000_(number)
Number used for counting
natural-number results: subtracting a larger natural number from a smaller one results in a negative number and dividing one natural number by another
Natural_number
Numbers parameterizing ways to partition a set
second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Integer having a non-trivial divisor
number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of
Composite_number
Matrix decomposition
orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for
QR_decomposition
Natural number
number following 44 and preceding 46.The number 45 is an odd composite number (3²×5), recognized as the 9th triangular number and a Kaprekar number.
45_(number)
Natural number, composite number
hexagonal lattice, 14 is also the number of fixed two-dimensional triangular-celled polyiamonds with four cells. 14 is the number of elements in a regular heptagon
14_(number)
Non-sinusoidal waveform
playing this file? See media help. A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise
Triangle_wave
Integer filtered out using a sieve similar to that of Eratosthenes
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the sieve of Eratosthenes
Lucky_number
Every positive integer is a sum of at most n n-gonal numbers
such representations of the number 17, for example, are shown below: 17 = 10 + 6 + 1 (triangular numbers) 17 = 16 + 1 (square numbers) 17 = 12 + 5 (pentagonal
Fermat polygonal number theorem
Fermat_polygonal_number_theorem
Number that represents a hexagon with a dot in the center
{n(n-1)}{2}}\right)} shows that the centered hexagonal number for n is 1 more than 6 times the (n − 1)th triangular number. In the opposite direction, the index n corresponding
Centered_hexagonal_number
Type of figurate number
number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number,
Hexagonal_number
Numbers obtained by adding the two previous ones
The only triangular Fibonacci numbers are 1, 3, 21, and 55, which was conjectured by Vern Hoggatt and proved by Luo Ming. No Fibonacci number can be a
Fibonacci_sequence
Recursive integer sequence
they were previously discovered in the 1730s by Minggatu. The n-th Catalan number can be expressed directly in terms of the central binomial coefficients
Catalan_number
Natural number
largest triangular number that is also a repdigit. Since 36 is a triangular number too, 666 is a doubly triangular number. Also, 666 is the sum of squares of
666_(number)
Two joined triangular cupolae
geometry, the triangular orthobicupola is the 27th Johnson solid. As the name suggests, it can be constructed by attaching two triangular cupolae along
Triangular_orthobicupola
centered square number, Mertens function zero 1014 = 210-10, Mertens function zero, sum of the nontriangular numbers between successive triangular numbers
1000_(number)
Base-dependent property of integers
mathematics, a natural number in a given number base is a p {\displaystyle p} -Kaprekar number if the representation of its square in that base can be split
Kaprekar_number
Natural number
number. a centered triangular number. a centered square number. a decagonal number. the smallest number that can be expressed as a sum of two squares
85_(number)
Natural number
both its prime factors being Gaussian primes. While 21 is the sixth triangular number, it is also the sum of the divisors of the first five positive integers:
21_(number)
Natural number
(twenty-eight) is the natural number following 27 and preceding 29. 28 is a composite number, a happy number, and a perfect number. 28 also appears in the Padovan
28_(number)
Natural number
J. A. (ed.). "Sequence A000537 (Sum of first n cubes; or n-th triangular number squared)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
100
Ten raised to an integer power
the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one
Power_of_10
Number that is the result of operation on its own digits
A Friedman number is an integer, which represented in a given numeral system, is the result of a non-trivial expression using all its own digits in combination
Friedman_number
Natural number
divides the Euclid number 2999# + 1 3003 – triangular number, only number known to appear eight times in Pascal's triangle; no number is known to appear
3000_(number)
Number that is less than the sum of its proper divisors
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The
Abundant_number
Constant used in a magic square
magic square which is also a: triangular number is 15 (solve the Diophantine equation x2 = y3 + 16y + 16, where y is divisible by 4); square number is 1
Magic_constant
Numbers with a certain property involving recursive summation
In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance
Happy_number
Type of natural number
In number theory, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors
Colossally_abundant_number
Numbers k where x - phi(x) = k has many solutions
In number theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions
Highly_cototient_number
Numeral ambigram
A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated by 180 degrees. In other words,
Strobogrammatic_number
Count of permutations by cycles
second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Natural number
ways: it is the 5th triangular number, a hexagonal number, and pentadecagonal number. a centered tetrahedral number. the smallest number that can be factorized
15_(number)
Integer divisible by sum of its digits
In recreational mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written
Harshad_number
Natural number
(four thousand) is the natural number following 3999 and preceding 4001. It is a decagonal number. 4005 – triangular number 4007 – safe prime 4010 – magic
4000_(number)
Class of natural numbers with many divisors
the number of divisors of n. The term was coined by Ramanujan (1915). For example, the number with the most divisors per square root of the number itself
Superior highly composite number
Superior_highly_composite_number
Natural number
triangle (as all triangular and tetragonal numbers appear in it). Because 15 is also triangular, 120 is a doubly triangular number. 120 is divisible
120_(number)
Concept in combinatorics
In mathematics, the cake number, denoted by Cn, is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly
Cake_number
Arithmetic operation
"possessions", "property") for a square—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area
Exponentiation
Triangular array of natural numbers
{\displaystyle \operatorname {N} (n,k),n\in \mathbb {N} ^{+},1\leq k\leq n} form a triangular array of natural numbers, called the Narayana triangle, that occur in
Narayana_number
Number whose divisors summed twice over equal twice itself
are any odd superperfect numbers. An odd superperfect number n would have to be a square number such that either n or σ(n) is divisible by at least three
Superperfect_number
Type of figurate number
A nonagonal number, or an enneagonal number, is a figurate number that extends the concept of triangular and square numbers to the nonagon (a nine-sided
Nonagonal_number
Type of number introduced by Mike Keith
mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number n {\displaystyle n} in a given number base b {\displaystyle
Keith_number
Prime number of the form 2^n – 1
mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer
Mersenne_prime
Combinatorial sequence of numbers
Dedekind number M ( n ) {\displaystyle M(n)} is the number of monotone Boolean functions of n {\displaystyle n} variables. Equivalently, it is the number of
Dedekind_number
Numbers that contain only the digit 1
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands
Repunit
Area of a right triangle with rational-numbered sides
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition
Congruent_number
Positive integer of the form (2^(2^n))+1
there infinitely many composite Fermat numbers? Does a Fermat number exist that is not square-free? As of December 2025[update], it is known that Fn is composite
Fermat_number
Mathematical sequences in combinatorics
matrix multiplications work because these matrices are lower triangular, so only a finite number of terms in the sum are nonzero. The Lah numbers L ( n ,
Stirling_number
Number of unique ways to draw non-intersecting chords in a circle
In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily
Motzkin_number
Type of Poulet number
In number theory, a super-Poulet number is a Poulet number, or pseudoprime to base 2, whose every divisor d {\displaystyle d} divides 2 d − 2 {\displaystyle
Super-Poulet_number
Number, non-palindrome after repeated sum with reverse
numbers exist? More unsolved problems in mathematics A Lychrel number is a natural number that cannot form a palindrome through the iterative process of
Lychrel_number
Abundant number whose proper divisors are all deficient numbers
primitive abundant number is an abundant number whose proper divisors are all deficient numbers. For example, 20 is a primitive abundant number because: The
Primitive_abundant_number
Composite number in number theory
In number theory, a Carmichael number is a composite number n {\displaystyle n} which in modular arithmetic satisfies the congruence relation: b n
Carmichael_number
Numbers that evenly divide powers of 60
calculation of square roots, such as how the Babylonians found an approximation to the square root of 2, perhaps using regular number approximations of
Regular_number
Number whose sums of distinct divisors represent all smaller numbers
In number theory, a practical number or panarithmic number is a positive integer n {\displaystyle n} such that all smaller positive integers can be represented
Practical_number
Number that has more digits than the number of digits in its prime factorization
In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization
Frugal_number
Number equal to the sum of all or some of its divisors
In number theory, a semiperfect number or pseudoperfect number is a natural number n equal to the sum of all or some of its proper divisors. A semiperfect
Semiperfect_number
Integer where the average of its positive divisors is also an integer
In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic
Arithmetic_number
Square root of the mean square
the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated
Root_mean_square
Integer named after Reo Fortune
(Fortune's conjecture) More unsolved problems in mathematics In number theory, a Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such
Fortunate_number
Number raised to the third power
proofs. For example, the sum of the first 5 cubes is the square of the 5th triangular number, 1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 15 2 {\displaystyle
Cube_(algebra)
SQUARED TRIANGULAR-NUMBER
SQUARED TRIANGULAR-NUMBER
Boy/Male
Arabic, Muslim
Scared
Boy/Male
African, American, Anglo, Australian, British, Christian, English, Jamaican
Battlefield; Spear Field; Triangular Field
Boy/Male
English
Lives in the triangular farm stead.
Boy/Male
American, British, English
Battlefield; From the Triangular Field
Boy/Male
Hindu, Indian
Scared
Boy/Male
English American
Shieldbearer.
Girl/Female
Muslim
Scared
Boy/Male
English
From the triangular field.
Girl/Female
Tamil
Scared
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Girl/Female
Hindu
Equaled, Similar
Boy/Male
Tamil
Harshnil | ஹரà¯à®·à¯à®¨à¯€à®²
Scared
Harshnil | ஹரà¯à®·à¯à®¨à¯€à®²
Boy/Male
Hindu, Indian, Marathi
Scared
Girl/Female
Hindu
Scared
Surname or Lastname
English
English : patronymic from Squire.
Boy/Male
American, Australian, British, English
Shield Bearer; Knight's Companion
Girl/Female
Tamil
Equaled, Similar
Boy/Male
French Latin
A squire.
Boy/Male
American, British, English
Lives in the Triangular Farm Stead
Boy/Male
Italian
Squire.
SQUARED TRIANGULAR-NUMBER
SQUARED TRIANGULAR-NUMBER
Boy/Male
British, English, Irish
Rambunctious; Boisterous; High-spirited; Variant of Shaun from John
Boy/Male
Muslim
Slave of the helper
Girl/Female
Australian, British, Christian, English, German, Indian, Latin
Garden of Plants and Flowers; Flower Name
Boy/Male
Tamil
Lord Ganesh, Soldier, Many
Boy/Male
Indian
Old Arabic name
Boy/Male
Bengali, Hindu, Indian
Lord Shib
Girl/Female
Tamil
Pranamya | பà¯à®°à®¾à®£à®®à®¯
Offering obeisances
Boy/Male
Hindu, Indian, Traditional
Victorious King
Male
Finnish
Finnish form of German Fridric, FREDRIIK means "peaceful ruler."Â
Girl/Female
Muslim/Islamic
A gift
SQUARED TRIANGULAR-NUMBER
SQUARED TRIANGULAR-NUMBER
SQUARED TRIANGULAR-NUMBER
SQUARED TRIANGULAR-NUMBER
SQUARED TRIANGULAR-NUMBER
a.
Forming a right angle; as, a square corner.
n. pl.
The triangular, or maioid, crabs. See Illust. under Maioid, and Illust. of Spider crab, under Spider.
n.
The product of a number or quantity multiplied by itself; thus, 64 is the square of 8, for 8 / 8 = 64; the square of a + b is a2 + 2ab + b2.
imp. & p. p.
of Squire
v. t.
To make triangular, or three-cornered.
a.
Oblong or elongated, and having three lateral angles; as, a triangular seed, leaf, or stem.
imp. & p. p.
of Square
a.
Rendering equal justice; exact; fair; honest, as square dealing.
adv.
In a triangular manner; in the form of a triangle.
n.
One who, or that which, squares.
n.
To multiply by itself; as, to square a number or a quantity.
n.
An instrument having at least one right angle and two or more straight edges, used to lay out or test square work. It is of several forms, as the T square, the carpenter's square, the try-square., etc.
n.
A square piece or fragment.
a.
Of or pertaining to a square, or to squares; resembling a quadrate, or square; square.
n.
A square; a measure; a rule.
adv.
In a square form or manner.
n.
A square. See 1st Squire.
n.
One who squares, or quarrels; a hot-headed, contentious fellow.
n.
Having the toe square.
n.
Hence, anything which is square, or nearly so