

A000027


The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.
(Formerly M0472 N0173)


1120



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

1,2


COMMENTS

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, 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, Jan 09 2005
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
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 1space defined by n points), A000124 (number of regions in 2space defined by n straight lines), A000125 (number of regions in 3space defined by n planes), A000127 (number of regions in 4space 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 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(n1) + 1. This has the following geometrical interpretation: Suppose there are already n1 points in general arrangement, thus defining the maximal number of regions on a straight line obtainable by n1 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(n1)=(n1)+1=n regions already there, hence a(n)=a(n1)+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, 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
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 NaturforscherVersammlung 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 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 theologicophilosophical statement.
Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 531. (End)
Binomial transform of A019590, inverse binomial transform of A001792.  Philippe Deléham, Oct 24 2007
Writing A000027 as N, perhaps the simplest onetoone 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(J1)/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 firstquadrant lattice by successive antidiagonals.  Clark Kimberling, Sep 11 2008
A000007(a(n)) = 0; A057427(a(n)) = 1.  Reinhard Zumkeller, Oct 12 2008
a(n) is also the mean of the first n odd integers.  Ian Kent, Dec 23 2008
Equals INVERTi transform of A001906, the evenindexed Fibonacci numbers starting (1, 3, 8, 21, 55,...).  Gary W. Adamson, Jun 05 2009
These are also the 2rough 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(p1) + 1 for prime p.  Jaroslav Krizek, Oct 18 2009
Triangle T(k,j) of natural numbers, read by rows, with T(k,j)=C(k,2)+j=.5(k^2k)+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=nC(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 nth triangular number.  Dennis P. Walsh, Nov 19 2009
HofstadterConwaylike sequence (see A004001): a(n) = a(a(n1)) + a(na(n1)) 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
Floyd's triangle read by rows.  Paul Muljadi, Jan 25 2010
Number of numbers between k and 2k where k is an integer.  Giovanni Teofilatto, Mar 26 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 in A178568.  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 ndigit numbers the binary expansion of which contains one run of 1's.  Vladimir Shevelev, Jul 30 2010
Let T denote the "natural number array A000027":
1....2.....4....7...
3....5.....8...12...
6....9....13...18...
10...14...19...25...
T(n,k)=n+(n+k2)(n+k1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and subarrays.  Clark Kimberling, Jan 29 2011
The Stern polynomial B(n,x) evaluated at x=2. See A125184.  T. D. Noe, Feb 28 2011
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 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
Number of partitions of 2n+1 into exactly two parts.  Wesley Ivan Hurt, Jul 15 2013
Integers n dividing u(n) = 2u(n1)  u(n2); u(0)=0, u(1)=1 (Lucas sequence A001477).  Thomas M. Bridge, Nov 03 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/(e2) = A194807.  Stanislav Sykora, Jan 20 2014
Engel expansion of e1 (A091131 = 1.71828...).  Jaroslav Krizek, Jan 23 2014
a(n) is the number of permutations of length n simultaneously avoiding 213, 231 and 321 in the classical sense which are breadthfirst search reading words of increasing unarybinary trees. For more details, see the entry for permutations avoiding 231 at A245898.  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 2n1 nodes. See A245904 for more information on increasing strict binary trees.  Manda Riehl Aug 07 2014


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 1.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, SpringerVerlag, 1990, page 25.
W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [From Leonid 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).


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..500000 [a large file]
Archimedes Laboratory, What's special about this number?
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
James Barton, The Numbers
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.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 371
Robert R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 2729.
E. Friedman, What's Special About This Number?
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Kival Ngaokrajang, Illustration about relation to many other sequences, when the sequence is considered as a triangular table read by its antidiagonals. Additional illustrations when the sequence is considered as a centered triangular table read by rows.
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 thirtyone
Uncyclopedia, Complete list of numbers from 1 to 20
G. Villemin's Almanac of Numbers, NOMBRES en BREF (in French)
Eric Weisstein's World of Mathematics, Natural Number, Positive Integer, Counting Number Composition, DavenportSchinzel Sequence, Idempotent Number, N, Smarandache Ceil Function, Whole Number, Engel Expansion, Trinomial Coefficient.
Wikipedia, List of numbers.
Wikipedia, Interesting number paradox.
Wikipedia, Floyd's triangle
Robert G. Wilson v, English names for the numbers from 0 to 11159 without spaces or hyphens .
Robert G. Wilson v, American English names for the numbers from 0 to 100999 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 partitioncounting sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,1).
Index to divisibility sequences


FORMULA

Multiplicative with a(p^e) = p^e.  David W. Wilson, Aug 01 2001
Another g.f.: Sum_{n>0} phi(n)x^n/(1x^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(s1).  Franklin T. AdamsWatters, Sep 11 2005
G.f.: x/(1x)^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. of A000108.  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
Convolution of A000012 (the allones sequence) with itself.  Tanya Khovanova, Jun 22 2007
a(n)=2*a(n1)a(n2); a(1)=1, a(2)=2. a(n)=1+a(n1).  Philippe Deléham, Nov 03 2008
a(n) = A000720(A000040(n)).  JuriStepan Gerasimov, Nov 29 2009
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k).  Philippe Deléham, Feb 10 2012
a(n) = Sum_{d  n} phi(d) = Sum_{d  n} A000010(d).  Jaroslav Krizek, Apr 20 2012
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(C(i+1,j), 1 <= i,j <= n), where C(n,k) are binomial coefficients.  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
From Wolfdieter Lang, Oct 09 2013: (Start)
a(n) = product(2*sin(Pi*k/n), k=1..n1), n >= 1.
a(n) = product((2*sin(Pi*k/(2*l)))^2, k=1..l1), l >= 1.
These identities are used in the calculation of products of length ratios of certain lines in a regular ngon. For the first identity see the GradsteinRyshik reference, p. 62, 1.392 1., bringing the first factor there to the left hand side and taking the limit x > 0 (L'Hospital). The second line follows from the first one. Thanks to Seppo Mustonen who led me to consider ngon lengths products. (End)
a(n) = sum(j=0..k, (1)^(j1)*j*C(n,j)*C(n1+kj,kj)), k>=0.  Mircea Merca, Jan 25 2014
a(n) = A052410(n) ^ A052409(n).  Reinhard Zumkeller, Apr 06 2014
a(n) = sum(k=1..n^2+2*n, 1/(sqrt(k)+sqrt(k+1))).  Pierre CAMI, Apr 25 2014


MAPLE

A000027 := n>n; seq(A000027(n), n=1..100);


MATHEMATICA

Range[100] (* Joseph Biberstine, Dec 26 2006 *)
t[n_, k_]:=n+(k+n2)(k+n1)/2; TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]] (* the array A000027 read by antidiagonals  Clark Kimberling, Jan 29 2011 *)
LinearRecurrence[{2, 1}, {1, 2}, 77] (* Robert G. Wilson v, May 22 2013 *)
CoefficientList[ Series[1/(x  1)^2, {x, 0, 76}], x] (* Robert G. Wilson v, May 22 2013 *)
NestList[# + 1 &, {1}, 76] // Flatten (* Robert G. Wilson v, May 13 2014 *)


PROG

(MAGMA) [ n : n in [1..100]];
(PARI) {a(n) = n};
(R) 1:100
(Shell) seq 1 100
(Haskell)
a000027 = id
a000027_list = [1..]  Reinhard Zumkeller, May 07 2012
(Maxima) makelist(n, n, 1, 30); /* Martin Ettl, Nov 07 2012 */


CROSSREFS

a(2k+1) = A005408(k), k >= 0, a(2k) = A005843(k), k >= 1.
Partial sums of A000012.
Cf. A001478, A007931, A007932, A001906, A178568, A027641, A074909, A001477.
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).
Cf. Boustrophedon transforms: A000737, A231179;
Cf. A038722 (mirrored when seen as triangle), A056011 (boustrophedon);
Cf. A194807.
Sequence in context: A131738 * A001477 A087156 A033619 A130734 A090108
Adjacent sequences: A000024 A000025 A000026 * A000028 A000029 A000030


KEYWORD

core,nonn,easy,mult,tabl,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

Links edited by Daniel Forgues, Oct 07 2009
Incorrect comment removed by Joerg Arndt, Mar 11 2010


STATUS

approved



