1,2

a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).

Inverse Euler transform of A000219.

The rectangular array having A000027 as 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 of A038722. - Clark Kimberling (ck6(AT)evansville.edu), Apr 05 2003

For nonzero x, define f(n)=floor(nx)-floor(n/x). Then f=A000027 if and only if x=tau or x=-tau. - Clark Kimberling (ck6(AT)evansville.edu), Jan 09 2005

Sum of powers of 2 (A007088) or algebraic sum of powers of 3 (A112867, A112952). - Lekraj Beedassy (blekraj(AT)yahoo.com), Mar 24 2006

Numbers of form (2^i)*k for odd k [i.e. n = A006519(n)*A000265(n)]; Thus n corresponds uniquely to an ordered pair (i,k) where i=A007814,k=A000265 {with A007814(2n)=A001511(n),A007814(2n+1)=0 } - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 22 2006

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, A008862, and A008863, 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 lines 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 following 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 on A000124 for 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 (james.east(AT)latrobe.edu.au), 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 (james.east(AT)latrobe.edu.au), May 03 2007

Comment from Clark Kimberling (ck6(AT)evansville.edu), Jul 07 2007: (Start) "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.

It is not clear, nor important, whether the "ganzen Zahlen" means the whole numbers, A000027, or all the integers, A130472. What is more important is that the adjective "liebe" in "liebe Gott." Walter Felscher explains that because "lieber Gott" is a colloquial phrase usually used only when speaking to children or illiterati, Kronecker's witticism was not intended as a theologico-philosophical statement.

Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. (End)

Binomial transform of A019590, inverse binomial transform of A001792 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007

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.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Robert R. Forslund, A Logical Alternative to the Existing Positional Number System, Southwest Journal of Pure and Applied Mathematics. Vol. 1 1995 pp. 27-29.

N. J. A. Sloane, Table of n, a(n) for n = 1..500000

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Archimedes Laboratory, What's special about this number?

C. K. Caldwell, Prime Curios

Case & Abiessu, interesting number

S. Crandall, notes on interesting digital ephemera

O. Curtis, Interesting Numbers

Walter Felscher, Historia Matematica Mailing List Archive.

Robert R. Forslund, A Logical Alternative to the Existing Positional Number System

E. Friedman, What's Special About This Number?

M. Keith, All Numbers Are Interesting: A Constructive Approach

R. Munafo, Notable Properties of Specific Numbers

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

R. Phillips, Numbers from one to thirty-one

Uncyclopedia, Complete list of numbers from 1 to 20

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Positive Integer

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Engel Expansion

Eric Weisstein's World of Mathematics, Trinomial Coefficient

Wikipedia, List of numbers

Wikipedia, Interesting number paradox

Robert G. Wilson v, English names for the numbers from 0 to 11159 without spaces or hyphens .

Index entries for "core" sequences

Index entries for sequences of the a(a(n)) = 2n family

Index entries for sequences that are permutations of the natural numbers

Index entries for related partition-counting sequences

Robert G. Wilson v, American English names for the numbers from 0 to 100999 without spaces or hyphens.

Multiplicative with a(p^e) = p^e. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

Another g.f.: Sum_{n>0} phi(n)x^n/(1-x^n) (Apostol).

When seen as 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

Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005.

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 g.f. A000108.- Michael Somos Sep 04 2006

Convolution of A000012 (all ones sequence) with itself. - Tanya Khovanova (tanyakh(AT)yahoo.com), Jun 22 2007

A000027 := n->n;

[ seq(n, n=1..100) ];

a[n_] := n

Range[60] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006

(MAGMA) [ n : n in [1..100]];

(PARI) a(n)=n

(SHELL) seq 1 100

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(2k+1)= A005408(k), k >= 0, a(2k)=A005843(k), k >= 1. Cf. A001477.

Partial sums of A000012.

Cf. A001478, A007931, A007932.

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 alphabtical order of English reverse spelling, A005589, etc.

Adjacent sequences: A000024 A000025 A000026 this_sequence A000028 A000029 A000030

Sequence in context: A023443 A099570 A020725 this_sequence A001477 A033619 A130734

core,nonn,easy,mult,tabl

njas