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A000027
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The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.
(Formerly M0472 N0173)
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2091
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
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OFFSET
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1,2
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COMMENTS
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For some authors, the terms "natural numbers" and "counting numbers" include 0, i.e., refer to the nonnegative integersA001477;the term "whole numbers" frequently also designates the whole set of (signed) integersA001057.
a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf.A007378).
The rectangular array havingA000027as antidiagonals is the dispersion of the complement of the triangular numbers,A000217(which triangularly form column 1 of this array). The array is also the transpose ofA038722.-Clark Kimberling,Apr 05 2003
For nonzero x, define f(n) = floor(nx) - floor(n/x). Then f=A000027if and only if x=tau or x=-tau. -Clark Kimberling,Jan 09 2005
If the offset were changed to 0, we would have the following pattern: a(n)=binomial(n,0) + binomial(n,1) for the present sequence (number of regions in 1-space defined by n points),A000124(number of regions in 2-space defined by n straight lines),A000125(number of regions in 3-space defined by n planes),A000127(number of regions in 4-space defined by n hyperplanes),A006261,A008859,A008860,A008861,A008862andA008863,where the last six sequences are interpreted analogously and in each "... by n..." clause an offset of 0 has been assumed, resulting in a(0)=1 for all of them, which corresponds to the case of not cutting with a hyperplane at all and therefore having one region. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
Define a number of points on a straight line to be in general arrangement when no two points coincide. Then these are the numbers of regions defined by n points in general arrangement on a straight line, when an offset of 0 is assumed. For instance, a(0)=1, since using no point at all leaves one region. The sequence satisfies the recursion a(n) = a(n-1) + 1. This has the following geometrical interpretation: Suppose there are already n-1 points in general arrangement, thus defining the maximal number of regions on a straight line obtainable by n-1 points, and now one more point is added in general arrangement. Then it will coincide with no other point and act as a dividing wall thereby creating one new region in addition to the a(n-1)=(n-1)+1=n regions already there, hence a(n)=a(n-1)+1. Cf. the comments onA000124for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
The sequence a(n)=n (for n=1,2,3) and a(n)=n+1 (for n=4,5,...) gives to the rank (minimal cardinality of a generating set) for the semigroup I_n\S_n, where I_n and S_n denote the symmetric inverse semigroup and symmetric group on [n]. -James East,May 03 2007
The sequence a(n)=n (for n=1,2), a(n)=n+1 (for n=3) and a(n)=n+2 (for n=4,5,...) gives the rank (minimal cardinality of a generating set) for the semigroup PT_n\T_n, where PT_n and T_n denote the partial transformation semigroup and transformation semigroup on [n]. -James East,May 03 2007
"God made the integers; all else is the work of man." This famous quotation is a translation of "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk," spoken by Leopold Kronecker in a lecture at the Berliner Naturforscher-Versammlung in 1886. Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. -Clark Kimberling,Jul 07 2007
WritingA000027as N, perhaps the simplest one-to-one correspondence between N X N and N is this: f(m,n) = ((m+n)^2 - m - 3n + 2)/2. Its inverse is given by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 + sqrt(8k - 7))/2). Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills the first-quadrant lattice by successive antidiagonals. -Clark Kimberling,Sep 11 2008
a(n) is also the mean of the first n odd integers. -Ian Kent,Dec 23 2008
Equals INVERTi transform ofA001906,the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55,...). -Gary W. Adamson,Jun 05 2009
These are also the 2-rough numbers: positive integers that have no prime factors less than 2. -Michael B. Porter,Oct 08 2009
Totally multiplicative sequence with a(p) = p for prime p. Totally multiplicative sequence with a(p) = a(p-1) + 1 for prime p. -Jaroslav Krizek,Oct 18 2009
Triangle T(k,j) of natural numbers, read by rows, with T(k,j) = binomial(k,2) + j = (k^2-k)/2 + j where 1 <= j <= k. In other words, a(n) = n = binomial(k,2) + j where k is the largest integer such that binomial(k,2) < n and j = n - binomial(k,2). For example, T(4,1)=7, T(4,2)=8, T(4,3)=9, and T(4,4)=10. Note that T(n,n)=A000217(n), the n-th triangular number. -Dennis P. Walsh,Nov 19 2009
Hofstadter-Conway-like sequence (seeA004001): a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = 1, a(2) = 2. -Jaroslav Krizek,Dec 11 2009
a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). -Leonid Bedratyuk,Jan 04 2010
Generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1), r = 2; in an infinite set, row 2 of the array shown inA178568.-Gary W. Adamson,May 29 2010
1/n = continued fraction [n]. Let barover[n] = [n,n,n,...] = 1/k. Then k - 1/k = n. Example: [2,2,2,...] = (sqrt(2) - 1) = 1/k, with k = (sqrt(2) + 1). Then 2 = k - 1/k. -Gary W. Adamson,Jul 15 2010
Number of n-digit numbers the binary expansion of which contains one run of 1's. -Vladimir Shevelev,Jul 30 2010
Let T denote the "natural number arrayA000027":
1 2 4 7...
3 5 8 12...
6 9 13 18...
10 14 19 25...
T(n,k) = n+(n+k-2)*(n+k-1)/2. SeeA185787for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays. (End)
The denominator in the Maclaurin series of log(2), which is 1 - 1/2 + 1/3 - 1/4 +.... -Mohammad K. Azarian,Oct 13 2011
As a function of Bernoulli numbers B_n (cf.A027641:(1, -1/2, 1/6, 0, -1/30, 0, 1/42,...)): let V = a variant of B_n changing the (-1/2) to (1/2). Then triangleA074909(the beheaded Pascal's triangle) * [1, 1/2, 1/6, 0, -1/30,...] = the vector [1, 2, 3, 4, 5,...]. -Gary W. Adamson,Mar 05 2012
Number of partitions of 2n+1 into exactly two parts. -Wesley Ivan Hurt,Jul 15 2013
For this sequence, the generalized continued fraction a(1)+a(1)/(a(2)+a(2)/(a(3)+a(3)/(a(4)+...))), evaluates to 1/(e-2) =A194807.-Stanislav Sykora,Jan 20 2014
a(n) is the number of permutations of length n simultaneously avoiding 213, 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 atA245898.-Manda Riehl,Aug 05 2014
a(n) is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. SeeA245904for more information on increasing strict binary trees. -Manda Riehl,Aug 07 2014
a(n) = least k such that 2*Pi - Sum_{h=1..k} 1/(h^2 - h + 3/16) < 1/n. -Clark Kimberling,Sep 28 2014
a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 < 1/n. -Clark Kimberling,Oct 02 2014
Determinants of the spiral knots S(2,k,(1)). a(k) = det(S(2,k,(1))). These knots are also the torus knots T(2,k). -Ryan Stees,Dec 15 2014
As a function, the restriction of the identity map on the nonnegative integers {0,1,2,3...},A001477,to the positive integers {1,2,3,...}. -M. F. Hasler,Jan 18 2015
See alsoA131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2)... (i^k+ k) / k! takes integral values for all i>=0: For k=1,A131685(k)=1, which implies that this is a well defined integer sequence. -Alexander R. Povolotsky,Apr 24 2015
a(n) is the number of compositions of n+2 into n parts avoiding the part 2. -Milan Janjic,Jan 07 2016
Does not satisfy Benford's law [Berger-Hill, 2017] -N. J. A. Sloane,Feb 07 2017
For n >= 3, a(n)=n is the least area that can be obtained for an irregular octagon drawn in a square of n units side, whose sides are parallel to the axes, with 4 vertices that coincide with the 4 vertices of the square, and the 4 remaining vertices having integer coordinates. See Affaire de Logique link. -Michel Marcus,Apr 28 2018
a(n+1) is the order of rowmotion on a poset defined by a disjoint union of chains of length n. -Nick Mayers,Jun 08 2018
Number of 1's in n-th generation of 1-D Cellular Automata using Rules 50, 58, 114, 122, 178, 186, 206, 220, 238, 242, 250 or 252 in the Wolfram numbering scheme, started with a single 1. -Frank Hollstein,Mar 25 2019
(1, 2, 3, 4, 5,...) is the fourth INVERT transform of (1, -2, 3, -4, 5,...). -Gary W. Adamson,Jul 15 2019
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.
W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [FromLeonid Bedratyuk,Jan 04 2010]
I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall,Spiral knots,Missouri J. of Math. Sci., 22 (2010).
M. DeLong, M. Russell, and J. Schrock,Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m),Involve, Vol. 8 (2015), No. 3, 361-384.
Eric Weisstein's World of Mathematics,Natural Number,Positive Integer,Counting NumberComposition,Davenport-Schinzel Sequence,Idempotent Number,N,Smarandache Ceil Function,Whole Number,Engel Expansion,andTrinomial Coefficient
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FORMULA
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Another g.f.: Sum_{n>0} phi(n)*x^n/(1-x^n) (Apostol).
When seen as an array: T(k, n) = n+1 + (k+n)*(k+n+1)/2. Main diagonal is 2n*(n+1)+1 (A001844), antidiagonal sums are n*(n^2+1)/2 (A006003). -Ralf Stephan,Oct 17 2004
G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).
Series reversion of g.f. A(x) is x*C(-x)^2 where C(x) is the g.f. ofA000108.-Michael Somos,Sep 04 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 4*u*v. -Michael Somos,Oct 03 2006
a(n) = 2*a(n-1)-a(n-2); a(1)=1, a(2)=2. a(n) = 1+a(n-1). -Philippe Deléham,Nov 03 2008
G.f.: x * Product_{j>=0} (1+x^(2^j))^2 = x * (1+2*x+x^2) * (1+2*x^2+x^4) * (1+2*x^4+x^8) *... = x + 2x^2 + 3x^3 +.... -Gary W. Adamson,Jun 26 2012
a(n) = det(binomial(i+1,j), 1 <= i,j <= n). -Mircea Merca,Apr 06 2013
E.g.f.: x*E(0), where E(k) = 1 + 1/(x - x^3/(x^2 + (k+1)/E(k+1) )); (continued fraction). -Sergei N. Gladkovskii,Aug 03 2013
a(n) = Product_{k=1..n-1} 2*sin(Pi*k/n), n > 1.
a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n > 1.
These identities are used in the calculation of products of ratios of lengths of certain lines in a regular n-gon. For the first identity see the Gradstein-Ryshik reference, p. 62, 1.392 1., bringing the first factor there to the left hand side and taking the limit x -> 0 (L'Hôpital). The second line follows from the first one. Thanks toSeppo Mustonenwho led me to consider n-gon lengths products. (End)
a(n) = Sum_{j=0..k} (-1)^(j-1)*j*binomial(n,j)*binomial(n-1+k-j,k-j), k>=0. -Mircea Merca,Jan 25 2014
a(n) = Sum_{k=1..n^2+2*n} 1/(sqrt(k)+sqrt(k+1)). -Pierre CAMI,Apr 25 2014
a(n) = floor(1/sin(1/n)) = floor(cot(1/(n+1))) = ceiling(cot(1/n)). -Clark Kimberling,Oct 08 2014
a(n) = 1/(1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 +...). -Pierre CAMI,Jan 22 2015
a(n) = Sum_{m=0..n-1} Stirling1(n-1,m)*Bell(m+1), for n >= 1. This corresponds to Bell(m+1) = Sum_{k=0..m} Stirling2(m, k)*(k+1), for m >= 0, from the fact that Stirling2*Stirling1 = identity matrix. SeeA048993,A048994andA000110.-Wolfdieter Lang,Feb 03 2015
a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k*(2n-k). In addition, surprisingly, a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k^2*(2n-k)^2. -Charlie Marion,Jan 05 2016
G.f.: x/(1-x)^2 = (x * r(x) *r(x^3) * r(x^9) * r(x^27) *...), where r(x) = (1 + x + x^2)^2 = (1 + 2x + 3x^2 + 2x^3 + x^4). -Gary W. Adamson,Jan 11 2017
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MAPLE
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MATHEMATICA
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PROG
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(Magma) [ n: n in [1..100]];
(PARI) {a(n) = n};
(R) 1:100
(Shell) seq 1 100
(Haskell)
a000027 = id
(Maxima) makelist(n, n, 1, 30); /*Martin Ettl,Nov 07 2012 */
(Python)
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CROSSREFS
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Cf.A026081= integers in reverse alphabetical order in U.S. English,A107322= English name for number and its reverse have the same number of letters,A119796= zero through ten in alphabetical order of English reverse spelling,A005589,etc. Cf.A185787(includes a list of sequences based on the natural number arrayA000027).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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