%I A000027 M0472 N0173
%S A000027 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,
%T A000027 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,
%U A000027 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
%N A000027 The natural numbers. Also called the whole numbers, the counting numbers
or the positive integers.
%C A000027 a(n) is smallest positive integer which is consistent with sequence being
monotonically increasing and satisfying a(a(n)) = n (cf. A007378).
%C A000027 Inverse Euler transform of A000219.
%C A000027 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
%C A000027 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
%C A000027 Sum of powers of 2 (A007088) or algebraic sum of powers of 3 (A112867,
A112952). - Lekraj Beedassy (blekraj(AT)yahoo.com), Mar 24 2006
%C A000027 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
%C A000027 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 A000027 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 A000027 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 A000027 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 A000027 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.
%C A000027 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.
%C A000027 Possibly the first publication of the statement is in Heinrich Weber's
"Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. (End)
%C A000027 Binomial transform of A019590, inverse binomial transform of A001792
. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
%C A000027 Contribution from Clark Kimberling (ck6(AT)evansville.edu), Sep 11 2008:
(Start)
%C A000027 Writing A000027 as N, perhaps the simplest one-to-one correspondence
between
%C A000027 NxN and N is this: f(m,n)=((m+n)^2 - m - 3n + 2)/2. Its inverse is given
%C A000027 by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 +
sqrt(8k - 7)/2).
%C A000027 Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills
the
%C A000027 first-quadrant lattice by successive antidiagonals. (End)
%C A000027 A000007(a(n)) = 0; A057427(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 12 2008]
%C A000027 a(n) is also the mean of the first n odd integers. [From Ian Kent (abides(AT)bu.edu),
Dec 23 2008]
%C A000027 For all n (1,2,3,4,5,6,...,) if Y=n, A=n^2+2, X=n^2+1 we have the Pell's
equation X^2-A*Y^2=1. Example: n=1=Y, A=3, X=2, 2^2-3*1=1; n=2=Y,
A=6, X=5, 5^2-6*2^2=1; n=3=Y, A=11, X=10, 10^2-11*3^2=1, and so on.
[From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]
%C A000027 Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers
starting (1, 3, 8, 21, 55,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Jun 05 2009]
%C A000027 These are also the 2-rough numbers: positive integers that have no prime
factors less than 2. [From Michael Porter (michael_b_porter(AT)yahoo.com),
Oct 08 2009]
%C A000027 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
(jaroslav.krizek(AT)atlas.cz), Oct 18 2009]
%C A000027 The positive numbers with smallest single divisor. A000027 = A000040
U A018252 = A006093 U A072668. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 22 2009]
%C A000027 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 Walsh (dwalsh(AT)mtsu.edu),
Nov 19 2009]
%D A000027 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 1.
%D A000027 T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory,
Springer-Verlag, 1990, page 25.
%D A000027 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A000027 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 A000027 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000027 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A000027 N. J. A. Sloane, <a href="b000027.txt">Table of n, a(n) for n = 1..500000</
a> [a large file]
%H A000027 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A000027 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas
for Some Functions on Finite Sets</a>
%H A000027 Archimedes Laboratory, <a href="http://www.archimedes-lab.org/numbers/
Num1_200.html">What's special about this number?</a>
%H A000027 C. K. Caldwell, <a href="http://primes.utm.edu/curios">Prime Curios</
a>
%H A000027 Case & Abiessu, <a href="http://everything2.net/index.pl?node_id=17633&displaytype=printable&lastnode_id=1763\
3">interesting number</a>
%H A000027 S. Crandall, <a href="http://tingilinde.typepad.com/starstuff/2005/11/
significant_int.html">notes on interesting digital ephemera</a>
%H A000027 O. Curtis, <a href="http://users.pipeline.com.au/owen/Numbers.html">Interesting
Numbers</a>
%H A000027 Walter Felscher, <a href="http://sunsite.utk.edu/math_archives/.http/
hypermail/historia/may99/0210.html">Historia Matematica Mailing List
Archive.</a>
%H A000027 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 A000027 E. Friedman, <a href="http://www.stetson.edu/~efriedma/numbers.html">
What's Special About This Number?</a>
%H A000027 M. Keith, <a href="http://users.aol.com/s6sj7gt/interest.htm">All Numbers
Are Interesting: A Constructive Approach</a>
%H A000027 R. Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Notable
Properties of Specific Numbers</a>
%H A000027 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 A000027 R. Phillips, <a href="http://richardphillips.org.uk/number/Num1.htm">
Numbers from one to thirty-one</a>
%H A000027 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 A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
NaturalNumber.html">Natural Number</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PositiveInteger.html">Positive Integer</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CountingNumber.html">Counting Number</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Composition.html">Composition</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Davenport-SchinzelSequence.html">Davenport-Schinzel Sequence</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
IdempotentNumber.html">Idempotent Number</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
N.html">N</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SmarandacheCeilFunction.html">Smarandache Ceil Function</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
WholeNumber.html">Whole Number</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
EngelExpansion.html">Engel Expansion</a>.
%H A000027 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TrinomialCoefficient.html">Trinomial Coefficient</a>.
%H A000027 Wikipedia, <a href="http://en.wikipedia.org/wiki/List_of_numbers">List
of numbers</a>.
%H A000027 Wikipedia, <a href="http://en.wikipedia.org/wiki/Interesting_number_paradox">
Interesting number paradox</a>.
%H A000027 Robert G. Wilson v, <a href="a000027.txt">English names for the numbers
from 0 to 11159 without spaces or hyphens </a>.
%H A000027 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000027 <a href="Sindx_Aa.html#aan">Index entries for sequences of the a(a(n))
= 2n family</a>
%H A000027 <a href="Sindx_Per.html#IntegerPermutation">Index entries for sequences
that are permutations of the natural numbers</a>
%H A000027 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A000027 Robert G. Wilson v, <a href="a001477.txt">American English names for
the numbers from 0 to 100999 without spaces or hyphens</a>.
%H A000027 James Barton, <a href="http://www.virtuescience.com/number-list.html">
The Numbers</a> [From Lekraj Beedassy (blekraj(AT)yahoo.com), Oct
31 2008]
%H A000027 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/
aNombre/Nb0a50.htm">NOMBRES en BREF (in French)</a> [From Lekraj
Beedassy (blekraj(AT)yahoo.com), Jan 01 2009]
%H A000027 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
Publications/books.html">Analytic Combinatorics</a>, 2009; see page
371
%F A000027 Multiplicative with a(p^e) = p^e. - David W. Wilson (davidwwilson(AT)comcast.net),
Aug 01, 2001.
%F A000027 Another g.f.: Sum_{n>0} phi(n)x^n/(1-x^n) (Apostol).
%F A000027 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 A000027 Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters,
Sep 11 2005.
%F A000027 G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).
%F A000027 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 A000027 Convolution of A000012 (all ones sequence) with itself. - Tanya Khovanova
(tanyakh(AT)yahoo.com), Jun 22 2007
%F A000027 a(n)=2*a(n-1)-a(n-2); a(1)=1, a(2)=2. a(n)=1+a(n-1). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Nov 03 2008]
%p A000027 A000027 := n->n;
%p A000027 [ seq(n,n=1..100) ];
%t A000027 a[n_] := n
%t A000027 Range[100] (from Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26
2006)
%o A000027 (MAGMA) [ n : n in [1..100]];
%o A000027 (PARI) a(n)=n
%o A000027 (R) 1:100
%o A000027 (SHELL) seq 1 100
%o A000027 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
%Y A000027 a(2k+1)= A005408(k), k >= 0, a(2k)=A005843(k), k >= 1. Cf. A001477.
%Y A000027 Partial sums of A000012.
%Y A000027 Cf. A001478, A007931, A007932.
%Y A000027 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.
%Y A000027 A001906 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2009]
%Y A000027 Cf. A000040, A018252, A006093, A072668. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 22 2009]
%Y A000027 Sequence in context: A020725 A119972 A131738 this_sequence A001477 A087156
A033619
%Y A000027 Adjacent sequences: A000024 A000025 A000026 this_sequence A000028 A000029
A000030
%K A000027 core,nonn,easy,mult,tabl,new
%O A000027 1,2
%A A000027 N. J. A. Sloane (njas(AT)research.att.com).
%E A000027 Links edited by Daniel Forgues (squid(AT)zensearch.com), Oct 07 2009
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