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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
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
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
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
Type of composite integer
706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985. (sequence A006753 in the OEIS) W.L. McDaniel in 1987 proved that there are infinitely
Smith_number
Natural number
number, a rhombic matchstick number, a happy number, and a member of Aronson's sequence. There are exactly 208 five-bead necklaces drawn from a set of beads
208_(number)
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
Natural number
of Aronson's sequence. At most 226 different permutation patterns can occur within a single 9-element permutation. Sloane, N. J. A. (ed.). "Sequence A007770
226_(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
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
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
Natural number
one of the numbers in Aronson's sequence. Sloane, N. J. A. (ed.). "Sequence A000959". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane
237_(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
Number that is the result of operation on its own digits
2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, ... (sequence A036057 in the OEIS). Friedman numbers are named after Erich Friedman,
Friedman_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
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
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
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
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
Integer filtered out using a sieve similar to that of Eratosthenes
Sloane, N. J. A. (ed.). "Sequence A031157 (Numbers that are both lucky and prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Guy,
Lucky_number
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
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 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
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
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
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
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 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
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
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
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
Number with odd number of 1s in binary
), "Sequence A000069 (Odious numbers: numbers with an odd number of 1's in their binary expansion)", The On-Line Encyclopedia of Integer Sequences, OEIS
Odious_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
Class of natural numbers with many divisors
5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers
Superior highly composite number
Superior_highly_composite_number
Number equal to the sum of its proper divisors
function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also S {\displaystyle
Perfect_number
Concept in combinatorics
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 in the
Cake_number
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
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
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
Number with few prime factors
Prime Sorting related Pancake number Sorting number Natural language related Aronson's sequence Ban Graphemics related Strobogrammatic Mathematics portal
Almost_prime
Number with a half-integer abundancy index
4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, ... (sequence A159907 in the OEIS) 24 is a hemiperfect number because the sum of the
Hemiperfect_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
Sum of a number's digits
Encyclopedia of Integer Sequences. Borwein & Borwein (1992) uses the generating function of this integer sequence (and of the analogous sequence for binary digit
Digit_sum
Integer named after Reo Fortune
37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, ... (sequence A005235 in the OEIS). The Fortunate numbers sorted in numerical order with
Fortunate_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
Result of multiplying seven instances of a number
number by its fifth power, or the cube of a number by its fourth power. The sequence of seventh powers of integers is: 0, 1, 128, 2187, 16384, 78125, 279936
Seventh_power
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
Type of positive composite integer
Prime Sorting related Pancake number Sorting number Natural language related Aronson's sequence Ban Graphemics related Strobogrammatic Mathematics portal
Lucas–Carmichael_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
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
Number that has a perfect number of factors adding up to another perfect number
numbers: 12 and (2126)(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1) (sequence A081357 in the OEIS). The second of these has 76 decimal digits: 6,086
Sublime_number
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 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
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
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
Number that has more digits than the number of digits in its prime factorization
base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers (sequence A046759 in the OEIS). The first frugal number which is not a prime power
Frugal_number
Prime number of the form 2^n – 1
31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in
Mersenne_prime
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
Positive integer whose divisors have a harmonic mean that is an integer
numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS). Harmonic divisor numbers were introduced by Øystein
Harmonic_divisor_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)
Type of natural number in recreational number theory
10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, ... (sequence A072857 in the OEIS) The number of primes that can be obtained from the
Primeval_number
Number that remains the same when its digits are reversed
131, 151, ... (sequence A002385 in the OEIS). The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... (sequence A002779 in the
Palindromic_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
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
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
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
Mathematical sequences in combinatorics
three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three
Stirling_number
Probable prime that is composite
Prime Sorting related Pancake number Sorting number Natural language related Aronson's sequence Ban Graphemics related Strobogrammatic Mathematics portal
Pseudoprime
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
Number, non-palindrome after repeated sum with reverse
a sequence of the first 126 numbers (125 of them never reported before) that take exactly 261 steps to reach a 119-digit palindrome. This sequence started
Lychrel_number
Number that has fewer digits than the number of digits in its prime factorization
base 10, 4 = 22, 6 = 2×3, 8 = 23, and 9 = 32 are extravagant numbers (sequence A046760 in the OEIS). There are infinitely many extravagant numbers in
Extravagant_number
Class of series of figurate numbers, each having a central dot
values" Neil Sloane & Simon Plouffe (1995). The Encyclopedia of Integer Sequences. San Diego: Academic Press.{{cite book}}: CS1 maint: publisher location
Centered_polygonal_number
Figurate number based on the stella octangula
form n(2n2 − 1). The sequence of stella octangula numbers is 0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ... (sequence A007588 in the OEIS) Only
Stella_octangula_number
Size of a geometric arrangement of points
Prime Sorting related Pancake number Sorting number Natural language related Aronson's sequence Ban Graphemics related Strobogrammatic Mathematics portal
Figurate_number
Integers occurring in the coefficients of the Taylor series of 1/cosh t
In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e
Euler_numbers
Number of orderings allowing ties
{\displaystyle n} elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race. The
Ordered_Bell_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
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
Number equal to the sum of all or some of its divisors
first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (sequence A005835 in the OEIS) Every multiple of a semiperfect number is semiperfect
Semiperfect_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
Integer divisible by sum of its digits
Harshad number many times over. So-called Trans-harshad numerals are sequences of the digits 0-9 which, in every base which uses all their digits, represent
Harshad_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
Number sequence 3,0,2,3,2,5,5,7,10,...
mathematics, the Perrin numbers are a doubly infinite constant-recursive integer sequence with characteristic equation x3 = x + 1. The Perrin numbers, named after
Perrin_number
Number n where the highest prime factor of (n^2 + 1) is at least 2n
85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 97... (sequence A005528 in the OEIS). The complementary sequence (numbers below 100 that aren't Størmer) is only
Størmer_number
Combinatorial sequence of numbers
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number
Dedekind_number
Numeral ambigram
6969, 8008, 8118, 8698, 8888, 8968, 9006, 9116, 9696, 9886, 9966, … (sequence A000787 in the OEIS) The first few strobogrammatic primes are: 11, 101
Strobogrammatic_number
Figurate number
given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4..., producing the sequence: 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51,
Pentagonal_number
Integer that occurs often as a totient
576, 720, 1152, 1440 (sequence A097942 in the OEIS), with 2, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 (sequence A131934 in the OEIS)
Highly_totient_number
Product of two prime numbers
51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95 (sequence A001358 in the OEIS) Semiprimes that are not square numbers are called
Semiprime
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
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
Number equal to the product of the sum and product of its digits
{\displaystyle b} . In base 10, there are exactly four sum-product numbers (sequence A038369 in the OEIS): 0, 1, 135, and 144. Let n {\displaystyle n} be a
Sum-product_number
Number n where phi(m) is greater than phi(n) for all m greater than n
5460, 5610, 5670, 6090, 6930, 7140, 7350, 8190, 9240, 9660, 9870, ... (sequence A036913 in the OEIS). The concept was introduced by David Masser and Peter
Sparsely_totient_number
Infinite integer series where the next number is the sum of the two preceding it
Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the
Lucas_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
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
Number of close-packed spheres in an octahedron
octahedral numbers are: 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 (sequence A005900 in the OEIS). The octahedral numbers have a generating function
Octahedral_number
Number in the 5th cell of any row of Pascal's triangle
this kind are: 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 (sequence A000332 in the OEIS) Pentatope numbers belong to the class of figurate
Pentatope_number
Number of points in an octagonal arrangement
40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936 (sequence A000567 in the OEIS) The octagonal number for n can also be calculated
Octagonal_number
Repeated sum of a number's digits
{(b-1)}})&{\mbox{if}}\ n\neq 0.\end{cases}}} In base 10, the corresponding sequence is (sequence A010888 in the OEIS). The digital root is the value modulo ( b −
Digital_root
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