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%I M0472 N0173
%S 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,
%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U 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
%N The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.
%C a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).
%C Inverse Euler transform of A000219.
%C 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_, Apr 05 2003
%C 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_, Jan 09 2005
%C Sum of powers of 2 (A007088) or algebraic sum of powers of 3 (A112867, A112952). - _Lekraj Beedassy_, Mar 24 2006
%C 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_, Apr 22 2006
%C 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
%C 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
%C 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
%C 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
%C Comment from _Clark Kimberling_, 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.
%C 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 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.
%C Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. (End)
%C Binomial transform of A019590, inverse binomial transform of A001792 . - _Philippe DELEHAM_, Oct 24 2007
%C Contribution from _Clark Kimberling_, Sep 11 2008: (Start)
%C Writing A000027 as N, perhaps the simplest one-to-one correspondence between
%C NxN and N is this: f(m,n)=((m+n)^2 - m - 3n + 2)/2. Its inverse is given
%C by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 + sqrt(8k - 7))/2).
%C Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills the
%C first-quadrant lattice by successive antidiagonals. (End)
%C A000007(a(n)) = 0; A057427(a(n)) = 1. [From _Reinhard Zumkeller_, Oct 12 2008]
%C a(n) is also the mean of the first n odd integers. [From _Ian Kent_, Dec 23 2008]
%C Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55,...). [From _Gary W. Adamson_, Jun 05 2009]
%C These are also the 2-rough numbers: positive integers that have no prime factors less than 2. [From _Michael B. Porter_, Oct 08 2009]
%C Totally multiplicative sequence with a(p) = p for prime p. Totally multiplicative sequence with a(p) = a(p-1) + 1 for prime p. [From _Jaroslav Krizek_, Oct 18 2009]
%C Triangle T(k,j) of natural numbers, read by rows, with T(k,j)=C(k,2)+j=.5(k^2-k)+j where 1<=j<=k. In other words, a(n)=n=C(k,2)+j where k is the largest integer such that C(k,2)<n and j=n-C(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. [From _Dennis P. Walsh_, Nov 19 2009]
%C Hofstadter-Conway-like sequence (see A004001): a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = 1, a(2) = 2. [From _Jaroslav Krizek_, Dec 11 2009]
%C a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2) [From _Leonid Bedratyuk_, Jan 04 2010]
%C Floyd's triangle read by rows. [From _Paul Muljadi_, Jan 25 2010]
%C Number of numbers between k and 2k where k is an integer. [From _Giovanni Teofilatto_, Mar 26 2010]
%C Contribution from _Gary W. Adamson_, May 29 2010: (Start)
%C Generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1), r = 2; in an infinite
%C set, row 2 of the array shown in A178568. (End)
%C Contribution from _Gary W. Adamson_, Jul 15 2010: (Start)
%C 1/n = continued fraction [n].
%C Let barover[n] = [n,n,n,...] = 1/k. Then k - 1/k = n.
%C Example: [2,2,2,...] = (sqrt(2) - 1) = 1/k, with k =
%C (sqrt(2) + 1). Then 2 = k - 1/k. (End)
%C Number of n-digit numbers the binary expansion of which contains one run of 1's. [From _Vladimir Shevelev_, Jul 30 2010]
%C Contribution from _Clark Kimberling_, Jan 29 2011: (Start)
%C Let T denote the "natural number array A000027":
%C 1....2.....4....7...
%C 3....5.....8...12...
%C 6....9....13...18...
%C 10...14...19...25...
%C T(n,k)=n+(n+k-2)(n+k-1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays.
%C (End)
%C The Stern polynomial B(n,x) evaluated at x=2. See A125184. - T. D. Noe, Feb 28 2011
%C The denominator in the Maclaurin series of ln 2, which is 1 - 1/2 + 1/3 - 1/4 +.... - Mohammad K. Azarian, Oct 13 2011
%C 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 triangle A074909 (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
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.
%D T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.
%D Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D 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.
%D W. FULTON and J. HARRIS. Representation theory: a first course. (1991). page 149 [From _Leonid Bedratyuk_, Jan 04 2010]
%D R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A000027/b000027.txt">Table of n, a(n) for n = 1..500000</a> [a large file]
%H Archimedes Laboratory, <a href="http://www.archimedes-lab.org/numbers/Num1_200.html">What's special about this number?</a>
%H James Barton, <a href="http://www.virtuescience.com/number-list.html">The Numbers</a>
%H C. K. Caldwell, <a href="http://primes.utm.edu/curios">Prime Curios</a>
%H Case & Abiessu, <a href="http://everything2.net/index.pl?node_id=17633&displaytype=printable&lastnode_id=17633">interesting number</a>
%H S. Crandall, <a href="http://tingilinde.typepad.com/starstuff/2005/11/significant_int.html">notes on interesting digital ephemera</a>
%H O. Curtis, <a href="http://users.pipeline.com.au/owen/Numbers.html">Interesting Numbers</a>
%H Walter Felscher, <a href="http://sunsite.utk.edu/math_archives/.http/hypermail/historia/may99/0210.html">Historia Matematica Mailing List Archive.</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 371
%H Robert R. Forslund, <a href="http://www.emis.de/journals/SWJPAM/Vol1_1995/rrf01.ps">A Logical Alternative to the Existing Positional Number System</a>
%H E. Friedman, <a href="http://www.stetson.edu/~efriedma/numbers.html">What's Special About This Number?</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H M. Keith, <a href="http://users.aol.com/s6sj7gt/interest.htm">All Numbers Are Interesting: A Constructive Approach</a>
%H R. Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Notable Properties of Specific Numbers</a>
%H G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H R. Phillips, <a href="http://richardphillips.org.uk/number/Num1.htm">Numbers from one to thirty-one</a>
%H Uncyclopedia, <a href="http://uncyclopedia.org/wiki/Complete_list_of_Numbers_from_1_to_20">Complete list of numbers from 1 to 20</a>
%H G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/aNombre/Nb0a50.htm">NOMBRES en BREF (in French)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NaturalNumber.html">Natural Number</a>, <a href="http://mathworld.wolfram.com/PositiveInteger.html">Positive Integer</a>, <a href="http://mathworld.wolfram.com/CountingNumber.html">Counting Number</a> <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>, <a href="http://mathworld.wolfram.com/Davenport-SchinzelSequence.html">Davenport-Schinzel Sequence</a>, <a href="http://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>, <a href="http://mathworld.wolfram.com/N.html">N</a>, <a href="http://mathworld.wolfram.com/SmarandacheCeilFunction.html">Smarandache Ceil Function</a>, <a href="http://mathworld.wolfram.com/WholeNumber.html">Whole Number</a>, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/List_of_numbers">List of numbers</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Interesting_number_paradox">Interesting number paradox</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Floyd%27s_triangle">Floyd's triangle</a>
%H Robert G. Wilson v, <a href="/A000027/a000027.txt">English names for the numbers from 0 to 11159 without spaces or hyphens </a>.
%H Robert G. Wilson v, <a href="/A001477/a001477.txt">American English names for the numbers from 0 to 100999 without spaces or hyphens</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Aa#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (2,-1).
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F Multiplicative with a(p^e) = p^e. - _David W. Wilson_, Aug 01, 2001.
%F Another g.f.: Sum_{n>0} phi(n)x^n/(1-x^n) (Apostol).
%F 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
%F Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005.
%F G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).
%F Series reversion of g.f. A(x) is x*C(-x)^2 where C(x) is g.f. A000108.- Michael Somos Sep 04 2006
%F 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
%F Convolution of A000012 (the all-ones sequence) with itself. - _Tanya Khovanova_, Jun 22 2007
%F a(n)=2*a(n-1)-a(n-2); a(1)=1, a(2)=2. a(n)=1+a(n-1). [From _Philippe DELEHAM_, Nov 03 2008]
%F a(n)=A000720(A000040(n)). [From _Juri-Stepan Gerasimov_, Nov 29 2009]
%F a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k). - Philippe Deléham, Feb 10 2012
%F a(n) = Sum_{d | n} phi(d)= Sum_{d | n} A000010(d). - Jaroslav Krizek, Apr 20 2012
%F 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
%F a(n) = det(C(i+1,j), 1 <= i,j <= n), where C(n,k) are binomial coefficients. [_Mircea Merca_, Apr 06 2013]
%p A000027 := n->n;
%p [ seq(n,n=1..100) ];
%t Range[100] (* from Joseph Biberstine, Dec 26 2006 *)
%t t[n_,k_]:=n+(k+n-2)(k+n-1)/2; TableForm[Table[t[n,k],{n,1,10},{k,1,10}]] (* the array A000027 read by anti-diagonals - From Clark Kimberling, Jan 29 2011 *)
%t LinearRecurrence[{2, -1}, {1, 2}, 77] (* _Robert G. Wilson v_, May 22 2013 *)
%t CoefficientList[ Series[1/(x - 1)^2, {x, 0, 76}], x] (* _Robert G. Wilson v_, May 22 2013 *)
%o (MAGMA) [ n : n in [1..100]];
%o (PARI) a(n)=n
%o (R) 1:100
%o (SHELL) seq 1 100
%o (Haskell)
%o a000027 = id
%o a000027_list = [1..] -- _Reinhard Zumkeller_, May 07 2012
%o (Maxima) makelist(n, n, 1, 30); /* _Martin Ettl_, Nov 07 2012 */
%Y a(2k+1)= A005408(k), k >= 0, a(2k)=A005843(k), k >= 1.
%Y Partial sums of A000012.
%Y Cf. A001478, A007931, A007932, A001906, A178568, A027641, A074909, A001477.
%Y 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 array A000027.)
%K core,nonn,easy,mult,tabl
%O 1,2
%A _N. J. A. Sloane_.
%E Links edited by _Daniel Forgues_, Oct 07 2009
%E Removed incorrect comment. - _Joerg Arndt_, Mar 11 2010
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