Searches , social queries for SUM FREE-SEQUENCE
Search references for SUM FREE-SEQUENCE. Phrases containing SUM FREE-SEQUENCE
See searches and references containing SUM FREE-SEQUENCE!
Search references for SUM FREE-SEQUENCE. Phrases containing SUM FREE-SEQUENCE
See searches and references containing SUM FREE-SEQUENCE!SUM FREE-SEQUENCE
Sequence of numbers avoiding sums of subsets
In mathematics, a sum-free sequence is an increasing sequence of positive integers, a 1 , a 2 , a 3 , … , {\displaystyle a_{1},a_{2},a_{3},\ldots ,} such
Sum-free_sequence
is the sum of any subset of the previous ones. The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570. The sum of the
List_of_sums_of_reciprocals
Set disjoint from its sumset with itself
of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words, A is sum-free if the equation a + b = c {\displaystyle
Sum-free_set
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
Infinite binary sequence generated by repeated complementation and concatenation
sequence can be defined by: ∏ i = 0 ∞ ( 1 − x 2 i ) = ∑ j = 0 ∞ ( − 1 ) t j x j , {\displaystyle \prod _{i=0}^{\infty }\left(1-x^{2^{i}}\right)=\sum _{j=0}^{\infty
Thue–Morse_sequence
Finite or infinite ordered list of elements
numbers are a sequence for which each element is the sum of the previous two elements. The zeroth and first elements are 0 and 1, so the sequence is (0, 1
Sequence
Topics referred to by the same term
refer to: Sum (category theory), the generic concept of summation in mathematics Sum, the result of summation, the addition of a sequence of numbers
Sum
Mathematical recursive sequence
aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches
Aliquot_sequence
Topics referred to by the same term
of a sequence Direct product Cartesian product of sets Direct product of groups Semidirect product Product of group subsets Wreath product Free product
Product
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
Natural number
the sum of its proper divisors, is greater than itself. The number appears in the Padovan sequence, preceded by 86, 114, and 151 (it is the sum of the
200_(number)
Index of articles associated with the same name
mathematics, a formal sum, formal series, or formal linear combination may be: In group theory, an element of a free abelian group, a sum of finitely many
Formal_sum
Number equal to the sum of its proper divisors
8589869056, and 137438691328 (sequence A000396 in the OEIS). The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one
Perfect_number
Italo-Swiss mathematician
problems with some notable contribution in particular concerning sum-free sequences. Among other problems in elementary number theory, he is the author
Giuseppe_Melfi
Integer divisible by sum of its digits
sum of the number's digits will be the number itself, and every number regardless of base or digits is divisible by themselves. Although the sequence
Harshad_number
Formal power series
ordinary generating function of a sequence an is: G ( a n ; x ) = ∑ n = 0 ∞ a n x n . {\displaystyle G(a_{n};x)=\sum _{n=0}^{\infty }a_{n}x^{n}.} If an
Generating_function
Online database of integer sequences
smallest row sums, is an example of a sequence with offset 3, and A072171, "Number of stars of visual magnitude n." is an example of a sequence with offset
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Number without repeated prime factors
encoding ∑ n = 0 ∞ a n ⋅ 2 n . {\displaystyle \sum _{n=0}^{\infty }{a_{n}}\cdot 2^{n}.} The square-free number 42 has factorization 2 × 3 × 7, or as an
Square-free_integer
Natural number
the second fourth-power of a prime: 34. with an aliquot sum of 40; within an aliquot sequence of three composite numbers (81,40,50,43,1,0) to the Prime
81_(number)
Programming paradigm
Python code. A sequence of operations such as the following: def example(x): return baz(bar(foo(x))) ... can be written in point-free style as the composition
Tacit_programming
Natural number
the free dictionary. 91 is: the twenty-seventh distinct semiprime and the second of the form (7.q), where q is a higher prime. the aliquot sum of 91
91_(number)
Pair of integers related by their divisors
constitutes an aliquot sequence of period 2. A related concept is that of a perfect number, which is a number that equals the sum of its own proper divisors
Amicable_numbers
Natural number
A. (ed.). "Sequence A036469 (Partial sums of A000009 (partitions into distinct parts))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
400_(number)
Number that is less than the sum of its proper divisors
84, 88, 90, and 96 (sequence A005101 in the OEIS). For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is
Abundant_number
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08. Sloane, N. J. A. (ed.). "Sequence A007504 (Sum of the first n primes.)". The
100
Natural number
the Moser–de Bruijn sequence because its base-4 representation (1000014) contains only digits 0 and 1, or equivalently, it's a sum of distinct powers of
1000_(number)
Natural number
Mian–Chowla sequence, and a strictly non-palindromic number. It is the sum of seven consecutive primes (71 + 73 + 79 + 83 + 89 + 97 + 101) and the sum of nine
500_(number)
Natural number between 89 and 91
Retrieved 2022-11-01. Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)"
90_(number)
Positive integer that is an integer power of another positive integer
576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, ... (sequence A001597 in the OEIS) The sum of the reciprocals of the perfect powers p without duplicates
Perfect_power
Number that cannot be written as an aliquot sum
Springer Verlag, 2004 ISBN 0-387-20860-7; section B10. OEIS sequence A070015 (Least m such that sum of aliquot parts of m equals n or 0 if no such number exists)
Untouchable_number
Natural number
sum of four consecutive primes (167 + 173 + 179 + 181). 701 is a prime number, a Chen prime, an Eisenstein prime with no imaginary part, and the sum of
700_(number)
Natural number
Retrieved 2023-05-04. Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)"
32_(number)
Natural number
Retrieved 2024-02-09. Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers: a(n) is 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider
41_(number)
Natural number
form 3 × q, where q is a higher prime. 15 is a deficient number because the sum of the proper divisors of 15 is less than 15. The prime factors of 15, 3
15_(number)
Natural number
geometric mean is 1,000,000. The sum of all its divisors (including itself) is 2,499,694,822,171; therefore, its aliquot sum is 1,499,694,822,171, and its
1,000,000,000,000
Natural number
corresponding aliquot sequence either terminates or ends in a repeating cycle. It follows the same sequence as 276, since its aliquot sum is the same as 276
300_(number)
Methods in computational biology
In bioinformatics, alignment-free sequence analysis approaches to molecular sequence and structure data provide alternatives over alignment-based approaches
Alignment-free sequence analysis
Alignment-free_sequence_analysis
Natural number
proper divisors 1, 3 and 7, twenty-one has a prime aliquot sum of 11 within an aliquot sequence containing only one composite number (21, 11, 1, 0). 21 is
21_(number)
Natural number
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
Algebraic structure formed from a collection of algebraic structures
infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2
Direct_sum
Abundant number whose proper divisors are all deficient numbers
abundant number because: The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. The sums of the proper divisors of 1,
Primitive_abundant_number
Topics referred to by the same term
the aliquot parts of an integer Aliquot sequence, a sequence of integers in which each number is the aliquot sum of the previous number Aliquot stringing
Aliquot
Natural number
Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A031971 (Sum_{1..n} k^n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
1,000,000
Type of mathematical sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N {\displaystyle N} , its subsequence x 1 , … , x N {\displaystyle
Low-discrepancy_sequence
Natural number
an Achilles number smallest four digit eban number the sum of all the nban numbers (sequence A008537 in the OEIS) 2002 = 2 × 7 × 11 × 13. It is a palindromic
2000_(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
Numbers whose sum of divisors is twice the number plus 1
are 20, 104, 464, 650, 1952, 130304, 522752, ... (sequence A088831 in the OEIS). Numbers n whose sum of factors equals 2 n − 2 {\displaystyle 2n-2} are
Quasiperfect_number
Type of sequence in numerical analysis
construct a sequence xn in Is so that lim n → ∞ 1 n ∑ i = 1 n f ( x i ) = ∫ I s f {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}f(x_{i})=\int
Sobol_sequence
Infinite sequence in mathematics
Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths
Kolakoski_sequence
Natural number
divides the sum of all primes <= p)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A125001 (Non-insertable
100,000
Natural number
second twin-prime distinct semiprime pair. The aliquot sum of 35 is 13, within an aliquot sequence of only one composite number (35,13,1,0) to the Prime
35_(number)
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-24. Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds
138_(number)
Type of composite integer
the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the
Smith_number
Product of the first "n" prime numbers
{\displaystyle \sum _{p\,{\text{prime}}}{1 \over p\#}={1 \over 2}+{1 \over 6}+{1 \over 30}+\ldots =0{.}7052301717918\ldots } (sequence A064648 in the OEIS)
Primorial
Product of an integer with itself
+n^{2}={\frac {n(n+1)(2n+1)}{6}}.} The first values of these sums, the square pyramidal numbers, are: (sequence A000330 in the OEIS) 0, 1, 5, 14, 30, 55, 91, 140
Square_number
Natural number
29 is the tenth supersingular prime. In this sequence, 29 is the seventeenth indexed member, where the sum of the largest two members (203, 290) is 17
29_(number)
Numbers whose aliquot sums form a cyclic sequence
aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or
Sociable_number
Natural number
first comprising 33, 34, 35. with an aliquot sum of 33; itself a semiprime, within an aliquot sequence of five composite numbers (87,33,15,9,4,3,1,0)
87_(number)
Number that is abundant but not semiperfect
the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to
Weird_number
Ordered list of whole numbers
complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. Integer sequences that have
Integer_sequence
Natural number
following 49 and preceding 51. Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways. 50 is an unsigned Stirling
50_(number)
Open-source typesetting system
typesetting system and corresponding markup language. The Typst compiler is free software and is distributed under the Apache License 2.0 license. The system
Typst
Number of elements in a subset of a commutative group
direct sum of k indecomposable subgroups of ranks r 1 , r 2 , … , r k {\displaystyle r_{1},r_{2},\ldots ,r_{k}} .[citation needed] Thus the sequence of ranks
Rank_of_an_abelian_group
Number equal to the sum of all or some of its divisors
is a natural number n equal to the sum of all or some of its proper divisors. A semiperfect number equal to the sum of all its proper divisors is a perfect
Semiperfect_number
Short exact sequence of sheaves on projective space
to an ( n + 1 ) {\displaystyle (n+1)} -fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle
Euler_sequence
Class of natural numbers with many divisors
6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors
Superior highly composite number
Superior_highly_composite_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
Number that is more than the sum of its proper divisors
47, 49, 50, ... (sequence A005100 in the OEIS) As an example, consider the number 21. Its proper divisors are 1, 3 and 7, and their sum is 11. Because 11
Deficient_number
Natural number
Padovan sequence 4411 – centered heptagonal number 4421 – super-prime, alternating factorial 4422 – pronic number 4425 = 15 + 25 + 35 + 45 + 55 4438 – sum of
4000_(number)
Number, product of consecutive integers
2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... (sequence A002378 in the OEIS). Letting P n {\displaystyle P_{n}} denote the pronic
Pronic_number
Number divisible only by 1 and itself
conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many
Prime_number
Topics referred to by the same term
Multigraph (orthography), a sequence of letters that behaves as a unit and is not the sum of its parts Multigraph (programming), a sequence of characters that
Multigraph_(disambiguation)
Numbers parameterizing ways to partition a set
a_{n}=\sum _{k=0}^{n}k!\left\{{n \atop k}\right\}.} Below is a triangular array of values for the Stirling numbers of the second kind (sequence A048993
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Inverse of a finite difference
In the calculus of finite differences, the indefinite sum, or antidifference, of a function f {\displaystyle f} is a solution F {\displaystyle F} of the
Indefinite_sum
Operation that combines groups
plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two
Free_product
Number, sum of distinct powers of 4
the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of
Moser–de_Bruijn_sequence
Number with a half-integer abundancy index
17428320, 20427264, 91963648, 197064960, ... (sequence A159907 in the OEIS) 24 is a hemiperfect number because the sum of the divisors of 24 is 1 + 2 + 3 + 4
Hemiperfect_number
Arithmetic function related to the divisors of an integer
OEIS: A001065), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function. For example, σ0(12) is the number
Divisor_function
Natural number
). "Sequence A003617 (Smallest n-digit prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000013
100,000,000,000
Natural number
octangula number. It is a Perrin number, coming after 22, 29, 39 in the sequence (and the sum of the first two). It is a Størmer number, since the greatest prime
51_(number)
Numbers whose sum of divisors is twice the number minus 1
natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n)
Almost_perfect_number
Area of a right triangle with rational-numbered sides
definition includes all positive rational numbers with this property. The sequence of (integer) congruent numbers starts with 5, 6, 7, 13, 14, 15, 20, 21
Congruent_number
Natural number
of Integer Sequences. OEIS Foundation. Retrieved 9 October 2023. Sloane, N. J. A. (ed.). "Sequence A048242 (Numbers that are not the sum of two abundant
23_(number)
Statement in abstract algebra
{\displaystyle d_{i}=0} . Such factors, if any, occur at the end of the sequence. While the direct sum is uniquely determined by M, the isomorphism giving the decomposition
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Mathematical inequality relating inner products and norms
of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors
Cauchy–Schwarz_inequality
Number that has a perfect number of factors adding up to another perfect number
a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28. As of July
Sublime_number
Natural number
Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A001923 (a(n) = Sum_{k=1..n} k^k.)". The On-Line Encyclopedia of Integer Sequences. OEIS
10,000,000
Statement in number theory
criterion is a particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis. The criterion is named after
Li's_criterion
Discrete probability distribution
}}\sum _{k=0}^{\min(k_{1},k_{2})}{\binom {k_{1}}{k}}{\binom {k_{2}}{k}}k!\left({\frac {\lambda _{3}}{\lambda _{1}\lambda _{2}}}\right)^{k}} The free Poisson
Poisson_distribution
Integer which is the sum of its positive unitary divisors, not including itself
19\times 37\times 79\times 109\times 157\times 313} (sequence A002827 in the OEIS). The respective sums of their proper unitary divisors are as follows: 6
Unitary_perfect_number
Number whose divisors summed twice over equal twice itself
{\displaystyle \sigma ^{2}(n)=\sigma (\sigma (n))=2n\,,} where σ is the sum-of-divisors function. Superperfect numbers are not a generalization of perfect
Superperfect_number
Number whose sums of distinct divisors represent all smaller numbers
1 + ∑ i = 1 j d i . {\displaystyle 2n\leq 1+\sum _{i=1}^{j}d_{i}.} In other words, the ordered sequence of all divisors d 1 < d 2 < . . . < d j {\displaystyle
Practical_number
Natural number
two distinct previously occurring Ulam numbers in a sequence. 69 is a deficient number because the sum of its proper divisors (which excludes itself) is
69_(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
Same digit count as prime factorization
example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigital numbers (sequence A046758 in the OEIS). All prime numbers are equidigital numbers in any
Equidigital_number
Multiplicative function in number theory
{\displaystyle M(n)=-1+\sum _{a\in {\mathcal {F}}_{n}}e^{2\pi ia},} where F n {\displaystyle {\mathcal {F}}_{n}} is the Farey sequence of order n {\displaystyle
Möbius_function
Grammar model in linguistics
incorporate sequence-structure relationship they lack the scoring metrics that reveal a sequence structural potential A weighted context-free grammar (WCFG)
Probabilistic context-free grammar
Probabilistic_context-free_grammar
Numbers whose prime factors all divide the number more than once
Riemann zeta function, and ζ(3) is Apéry's constant. (sequence A082695 in the OEIS) More generally, the sum of the reciprocals of the sth powers of the powerful
Powerful_number
Type of composite number
derivative of n. (For square-free numbers n = ∏ i p i {\displaystyle n=\prod _{i}{p_{i}}} , n ′ = ∑ i n p i {\displaystyle n'=\sum _{i}{\frac {n}{p_{i}}}}
Giuga_number
Type of natural number
sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one
Colossally_abundant_number
travel, tourism, insurance
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
SUM FREE-SEQUENCE
travel, tourism, insurance