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Counts pieces of a disk cut by lines
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza
Lazy_caterer's_sequence
Concept in combinatorics
of the lazy caterer's sequence. The values of Cn for n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... (sequence A000125
Cake_number
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-28. "Sloane's A000124 : Central polygonal numbers (the Lazy Caterer's sequence)". The On-Line
154_(number)
Online database of integer sequences
the prime numbers, the palindromic primes, the Fibonacci sequence, the lazy caterer's sequence, and the coefficients in the series expansion of ζ ( n +
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Natural number
Retrieved 2026-09-20. Sloane, N. J. A. (ed.). "Sequence A000124 (Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces
301_(number)
Numbers obtained by adding the two previous ones
Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known
Fibonacci_sequence
Array of partial sums of the binomial coefficients
of Integer Sequences, 19 (2016) 16.8.3. Sloane, N. J. A. (ed.). "Sequence A000124 (Central polygonal numbers (the Lazy Caterer's sequence))". The On-Line
Bernoulli's_triangle
Iterative algorithm on numbers
-\beta } to produce the next number of the sequence. Repeat step 2. The sequence is called a Kaprekar sequence and the function K b ( n ) = α − β {\displaystyle
Kaprekar's_routine
Natural number
510511 = 2 × 3 × 5 × 7 × 11 × 13 × 17 + 1. As a member of the lazy caterer's sequence, 277 counts the maximum number of pieces obtained by slicing a
277_(number)
Number that cannot be written as an aliquot sum
324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... (sequence A005114 in the OEIS). Unsolved problem in mathematics Are there any odd
Untouchable_number
Number that when multiplied by another number moves its last digit to its front
numbers posed by Freeman Dyson. They are: (leading zeros are not allowed) (sequence A092697 in the OEIS) In general, if we relax the rules to allow a leading
Parasitic_number
Sequence of integers
In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values: P ( 0 ) = P ( 1 ) = P ( 2 ) = 1 , {\displaystyle
Padovan_sequence
Type of composite number with an even number of digits
124483, 125248, 125433, 125460, 125500, ... (sequence A014575 in the OEIS) There are many known sequences of infinitely many vampire numbers following
Vampire_number
Number divisible only by 1 and itself
19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (sequence A000040 in the OEIS). No even number n {\displaystyle n} greater than
Prime_number
Integer having a non-trivial divisor
15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36. (sequence A002808 in the OEIS) Every composite number can be written as the product
Composite_number
Sum of a number's digits
Encyclopedia of Integer Sequences. Borwein & Borwein (1992) use the generating function of this integer sequence (and of the analogous sequence for binary digit
Digit_sum
Positive integer that is the product of three distinct prime numbers
numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... (sequence A007304 in the OEIS) The largest known sphenic number at any time can be
Sphenic_number
Natural number
OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000124 (Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces
300_(number)
Probable prime that is composite
Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus
Pseudoprime
Natural number
J. A. (ed.). "Sequence A000124 (Central polygonal numbers (the Lazy Caterer's sequence))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
232_(number)
Figurate number
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral
Triangular_number
is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to OEIS
List_of_integer_sequences
Triangular array of consecutive natural numbers
constructs. The numbers along the left edge of the triangle are the lazy caterer's sequence and the numbers along the right edge are the triangular numbers
Floyd's_triangle
Centered figurate number
11353, and 11881. (sequence A003154 in the OEIS) The digital root of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two
Star_number
Odd number with specific properties
n − 1 {\displaystyle k\times 2^{n}-1} , then k is a Riesel number. The sequence of currently known Sierpiński numbers begins with: 78557, 271129, 271577
Sierpiński_number
Integer whose multiples are digit rotations
necessary structure given in the next section. Allowing leading zeros, the sequence of cyclic numbers begins: (106 − 1) / 7 = 142857 (6 digits) (1016 − 1)
Cyclic_number
Product of an integer with itself
\lfloor x\rfloor } represents the floor of the number x. The squares (sequence A000290 in the OEIS) smaller than 602 = 3600 are: 02 = 0 12 = 1 22 = 4
Square_number
Mathematical formula
8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0. (sequence A031347 in the OEIS) Multiplicative digital roots are the multiplicative
Multiplicative_digital_root
Power of a prime number
powers, while 6 = 2 × 3, 12 = 22 × 3 and 36 = 62 = 22 × 32 are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25
Prime_power
Numbers whose sum of divisors is twice the number plus 1
5^{2}\times 13} . They are 20, 104, 464, 650, 1952, 130304, 522752, ... (sequence A088831 in the OEIS). Numbers n whose sum of factors equals 2 n − 2 {\displaystyle
Quasiperfect_number
Number used for counting
objects "larger", than the other. A sequence is a list of objects in a specific order. More precisely, a sequence is a function that assigns an object
Natural_number
Numbers k where x - phi(x) = k has many solutions
509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, ... (sequence A100827 in the OEIS) Many of the highly cototient numbers are odd. The
Highly_cototient_number
Number that is more than the sum of its proper divisors
31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... (sequence A005100 in the OEIS) As an example, consider the number 21. Its proper
Deficient_number
Sequence of rational numbers
In mathematics, a Göbel sequence is a sequence of rational numbers defined by the recurrence relation x n = x 0 2 + x 1 2 + ⋯ + x n − 1 2 n − 1 , {\displaystyle
Göbel's_sequence
Natural number
member of the Padovan sequence, a Harshad number, and the 30th triangular number. 466 = 2 × 233. It is a noncototient and a lazy caterer number. 467 is a prime
400_(number)
Result of multiplying four instances of a number together
zenzizenzic, biquadrate or supercubed instead of "to the power of 4". The sequence of fourth powers of integers, known as biquadrates or tesseractic numbers
Fourth_power
Mathematical concept
1952, only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS). Note that these numbers are all prime numbers. The
Lucky_numbers_of_Euler
Property of a number
10, 19, 199, 19999999999999999999999, ... (sequence A006050 in the OEIS) The next number in the sequence (the smallest number of additive persistence
Persistence_of_a_number
Size of a geometric arrangement of points
Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus
Figurate_number
Type of natural number
1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, ... (sequence A006378 in the OEIS) (sequence A003052 in the OEIS) Sándor & Crstici (2004) p.384 Sándor
Self_number
Result of multiplying five instances of a number together
number by its fourth power, or the square of a number by its cube. The sequence of fifth powers of integers is: 0, 1, 32, 243, 1024, 3125, 7776, 16807
Fifth_power_(algebra)
Integer having only small prime factors
the positive divisors of “the least common multiple of 1, 2, 3, …, n” (sequence A003418 in the OEIS), e.g. the 9-powersmooth numbers (also the 10-powersmooth
Smooth_number
Product of the first "n" prime numbers
{\displaystyle p_{n}\#} are: 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... (sequence A002110 in the OEIS). Asymptotically, primorials grow according to p n
Primorial
Class of binary number
These numbers give the positions of the zero values in the Thue–Morse sequence, and for this reason they have also been called the Thue–Morse set. Non-negative
Evil_number
Type of Poulet number
and a super-Poulet number. The super-Poulet numbers below 10,000 are (sequence A050217 in the OEIS): It is relatively easy to get super-Poulet numbers
Super-Poulet_number
Count of permutations by cycles
Mathematica. Other software packages for guessing formulas for sequences (and polynomial sequence sums) involving Stirling numbers and other special triangles
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Integer of the form 3 × 2^n – 1 for non-negative n
6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... (sequence A055010 in the OEIS) The 9th century mathematician, physician, astronomer
Thabit_number
Number in combinatorics
an integer sequence that can be used to count the plane trees with a given set of leaves, the ways of inserting parentheses into a sequence, and the ways
Schröder–Hipparchus_number
Product of prime numbers, plus one
2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... (sequence A006862 in the OEIS). The first few Kummer numbers are 1, 5, 29, 209, 2309
Euclid_number
Figurate number
(sequence A000292 in the OEIS) The first few square pyramidal numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... (sequence A000330
Pyramidal_number
Positive integer that is an integer power of another positive integer
considered perfect powers (0k = 0 for any k > 0, 1k = 1 for any k). A sequence of perfect powers can be generated by iterating through the possible values
Perfect_power
Number of dots in a centred dot square
2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, … (sequence A001844 in the OEIS). Each centered square number is the sum of successive
Centered_square_number
Number of paths between grid corners, allowing diagonal steps
n ) {\displaystyle D(m,n)} also counts the global alignments of two sequences of lengths m {\displaystyle m} and n {\displaystyle n} , the points in
Delannoy_number
Numbers in a type of Lucas sequence
integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence U n
Jacobsthal_number
Numbers whose aliquot sums form a cyclic sequence
form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable
Sociable_number
Concatenation of the first n prime numbers
1033, 2297, 3037, 11927, ... (sequence A046284 in the OEIS). The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers
Smarandache–Wellin_number
Number equal to the sum of its proper divisors
numbers are 6, 28, 496, 8128, 33550336, 8589869056, and 137438691328 (sequence A000396 in the OEIS). The sum of proper divisors of a number is called
Perfect_number
Numbers whose prime factors all divide the number more than once
648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... (sequence A001694 in the OEIS). If m = a2b3, then every prime in the prime factorization
Powerful_number
Recursive integer sequence
The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named
Catalan_number
Mathematical integer sequence
first few Schröder numbers are 1, 2, 6, 22, 90, 394, 1806, 8558, ... (sequence A006318 in the OEIS). where S 0 = 1 {\displaystyle S_{0}=1} and S 1 = 2
Schröder_number
Type of number introduced by Mike Keith
True sequence = [] y = x while y > 0: sequence.append(y % b) y = y // b digit_count = len(sequence) sequence.reverse() while sequence[len(sequence) - 1]
Keith_number
Abundant number whose proper divisors are all deficient numbers
abundant numbers are: 20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS) The smallest odd primitive abundant number is 945
Primitive_abundant_number
Centered figurate number that represents an octagon with a dot in the center
J. A. (ed.). "Sequence A016754 (Odd squares: (2n-1)^2. Also centered octagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
Centered_octagonal_number
Ten raised to an integer power
ten are: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000... (sequence A011557 in the OEIS) In decimal notation the nth power of ten is written
Power_of_10
Numbers that contain only the digit 1
in base 10 representation. The sequence of repunits base-10 starts with 1, 11, 111, 1111, 11111, 111111, ... (sequence A002275 in the OEIS). Similarly
Repunit
Numbers with special prime factorization
3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 (sequence A052486 in the OEIS). The sequence grows as O(n2/log log n), and the sum of reciprocals
Achilles_number
Type of prime number conjectured to exist
known. Let p {\displaystyle p} be a prime number. When each term in the sequence of Fibonacci numbers F n {\displaystyle F_{n}} is reduced modulo p {\displaystyle
Wall–Sun–Sun_prime
Number of unique ways to draw non-intersecting chords in a circle
1 , … {\displaystyle n=0,1,\dots } form the sequence: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ... (sequence A001006 in the OEIS) The following figure shows
Motzkin_number
Arithmetic operation
alternating sequences. For a similar discussion of powers of the complex number i, see § nth roots of a complex number. The limit of a sequence of powers
Exponentiation
Integer that is both a perfect square and a triangular number
1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in the OEIS) Write N k {\displaystyle N_{k}} for the k {\displaystyle
Square_triangular_number
Numbers parameterizing ways to partition a set
triangular array of values for the Stirling numbers of the second kind (sequence A048993 in the OEIS): As with the binomial coefficients, this table could
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Polyhedral number representing a tetrahedron
tetrahedral numbers are: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... (sequence A000292 in the OEIS) The formula for the nth tetrahedral number is represented
Tetrahedral_number
Numbers with a certain property involving recursive summation
1^{2}+0^{2}=1} . On the other hand, 4 is not a happy number because the sequence starting with 4 2 = 16 {\displaystyle 4^{2}=16} and 1 2 + 6 2 = 37 {\displaystyle
Happy_number
Triangular array of natural numbers
k}{n \choose k-1}} The first eight rows of the Narayana triangle read: (sequence A001263 in the OEIS) An example of a counting problem whose solution can
Narayana_number
Positive integer with large prime factors
6 mod 8 nor == 3, 6 mod 9, etc. The On-Line Encyclopedia of Integer Sequences (OEIS) lists p-rough numbers for small p: 2-rough numbers: A000027 3-rough
Rough_number
Numbers that evenly divide powers of 60
45, 48, 50, 54, 60, ... (sequence A051037 in the OEIS) Several other sequences at the On-Line Encyclopedia of Integer Sequences have definitions involving
Regular_number
Number of the form x^y + y^x
numbers are 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 (sequence A076980 in the OEIS). The requirement that x and y both be greater than
Leyland_number
Composite number which passes Miller–Rabin primality test
53971, 79381, ... (sequence A020231 in the OEIS). For base 4, see (sequence A020230 in the OEIS), and for bases 6 to 100, see (sequence A020232 in the OEIS)
Strong_pseudoprime
Three raised to an integer power
powers of three are: 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, etc. (sequence A000244 in OEIS) The powers of three give the place values in the ternary
Power_of_three
Number whose first n digits is a multiple of n
204, 207, 222, 225, 228, 243, 246, 249, 261, 264, 267, 282, 285, 288... (sequence A144688 in the OEIS) The smallest base 10 polydivisible numbers with n
Polydivisible_number
Centered figurate number that represents a heptagon with a dot in the center
"Sequence A069099 (Centered heptagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A144974
Centered_heptagonal_number
Number that is less than the sum of its proper divisors
30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, and 96 (sequence A005101 in the OEIS). For example, the proper divisors of 24 are 1, 2,
Abundant_number
Mathematical concept
In mathematics, a Cullen number is a member of the integer sequence C n = n ⋅ 2 n + 1 {\displaystyle C_{n}=n\cdot 2^{n}+1} (where n {\displaystyle n} is
Cullen_number
Concept in number theory
1125, 1224, 1242, 1287, 1440, 1503, 1566, 1611, 1620, 1800, 1935, ... (sequence A248970 in the OEIS) There are only 88 narcissistic numbers in base 10
Narcissistic_number
Type of natural number
5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A004490 in the OEIS) are also the first 15 superior highly composite numbers
Colossally_abundant_number
Number of the form (n * 2^n) - 1
n. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS). Woodall numbers were first studied by Allan J. C
Woodall_number
Sequence of numbers
Aronson's sequence is an integer sequence defined by the English sentence "T is the first, fourth, eleventh, sixteenth, ... letter in this sentence."
Aronson's_sequence
Number whose divisors summed twice over equal twice itself
superperfect numbers are: 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS). To illustrate: it can be seen that 16 is a superperfect
Superperfect_number
Two raised to an integer power
non-negative values of n are: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (sequence A000079 in the OEIS) By comparison, powers of two with negative exponents
Power_of_two
Integer whose representation contains every digit in its number base
pandigital number in base 10. The first few pandigital base 10 numbers are (sequence A171102 in the OEIS): 1023456789, 1023456798, 1023456879, 1023456897, 1023456978
Pandigital_number
Count of the possible partitions of a set
203 , 877 , 4140 , … {\displaystyle 1,1,2,5,15,52,203,877,4140,\dots } (sequence A000110 in the OEIS). The Bell number B n {\displaystyle B_{n}} counts
Bell_number
Type of integer in number theory
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequence A138591 in the OEIS). The impolite numbers are exactly the powers of two
Polite_number
Class of natural numbers
few superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... (sequence A004394 in the OEIS). For example, the number 5 is not a superabundant
Superabundant_number
Natural number with a decimal representation made of repeated instances of the same digit
infinitely many non-Brazilian primes, forming the sequence 2, 3, 5, 11, 17, 19, 23, 29, 37, 41, 47, 53, ... (sequence A220627 in the OEIS) If a Fermat number F
Repdigit
Numbers with many divisors
The first 41 highly composite numbers are listed in the table below (sequence A002182 in the OEIS). The number of divisors is given in the column labeled
Highly_composite_number
Type of positive integer
following property: there exists a positive integer a such that in the sequence (a, a + 1, …, a + k) of consecutive integers, each of the elements has
Erdős–Woods_number
Integer divisible by the number of its divisors
_{i=1}^{n}p_{i}^{e_{i}}} . The first few refactorable numbers are listed in (sequence A033950 in the OEIS) as 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80
Refactorable_number
Product of two distinct primes ≡ 3 (mod 4)
321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ... (sequence A016105 in the OEIS) The integers were named for computer scientist Manuel
Blum_integer
Type of number
(sequence A054377 in the OEIS). The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence, but then
Primary_pseudoperfect_number
Integer where the average of its positive divisors is also an integer
and 2, and their average 3/2 is not an integer. The first numbers in the sequence of arithmetic numbers are 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21
Arithmetic_number
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LAZY CATERERS-SEQUENCE
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