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A000027
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The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.
(Formerly M0472 N0173)
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908
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 (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 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
Contribution from Clark Kimberling (ck6(AT)evansville.edu), Sep 11 2008: (Start)
Writing A000027 as N, perhaps the simplest one-to-one correspondence between
NxN 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. (End)
A000007(a(n)) = 0; A057427(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 12 2008]
a(n) is also the mean of the first n odd integers. [From Ian Kent (abides(AT)bu.edu), Dec 23 2008]
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]
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]
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]
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]
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 (jaroslav.krizek(AT)atlas.cz), Dec 11 2009]
a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2) [From Leonid Bedratyuk (leonid.uk(AT)gmail.com), Jan 04 2010]
Floyd's triangle read by rows. [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Jan 25 2010]
Number of numbers between k and 2k where k is a integer. [From Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Mar 26 2010]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 29 2010: (Start)
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. (End)
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 15 2010: (Start)
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. (End)
Number of n-digit numbers the binary expansion of which contains one run of 1's. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 30 2010]
Contribution from Clark Kimberling (ck6(AT)evansville.edu), Jan 29 2011: (Start)
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+k-2)(n+k-1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays.
(End)
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 ln 2, which is 1 - 1/2 + 1/3 - 1/4 +.... - Mohammad K. Azarian, Oct 13 2011
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REFERENCES
| 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.
W. FULTON and J. HARRIS. Representation theory: a first course. (1991). page 149 [From Leonid Bedratyuk (leonid.uk(AT)gmail.com), Jan 04 2010]
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
| N. J. A. Sloane, Table of n, a(n) for n = 1..500000 [a large file]
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Archimedes Laboratory, What's special about this number?
James Barton, The Numbers [From Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 31 2008]
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
E. Friedman, What's Special About This Number?
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
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
G. Villemin's Almanac of Numbers, NOMBRES en BREF (in French) [From Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 01 2009]
Eric Weisstein's World of Mathematics, Natural Number.
Eric Weisstein's World of Mathematics, Positive Integer.
Eric Weisstein's World of Mathematics, Counting Number.
Eric Weisstein's World of Mathematics, Composition.
Eric Weisstein's World of Mathematics, Davenport-Schinzel Sequence.
Eric Weisstein's World of Mathematics, Idempotent Number.
Eric Weisstein's World of Mathematics, N.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.
Eric Weisstein's World of Mathematics, Whole Number.
Eric Weisstein's World of Mathematics, Engel Expansion.
Eric Weisstein's World of Mathematics, Trinomial Coefficient.
Wikipedia, List of numbers.
Wikipedia, Interesting number paradox.
Wikipedia, Floyd's triangle [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Jan 25 2010]
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 partition-counting sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,-1).
Index to divisibility sequences
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FORMULA
| 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 (the all-ones sequence) with itself. - Tanya Khovanova (tanyakh(AT)yahoo.com), Jun 22 2007
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]
a(n)=A000720(A000040(n)). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 29 2009]
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k). - DELEHAM Philippe, Feb 10 2012
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MAPLE
| A000027 := n->n;
[ seq(n, n=1..100) ];
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MATHEMATICA
| Range[100] (* from Joseph Biberstine, Dec 26 2006 *)
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 - From Clark Kimberling, Jan 29 2011 *)
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PROG
| (MAGMA) [ n : n in [1..100]];
(PARI) a(n)=n
(R) 1:100
(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
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CROSSREFS
| 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.
Cf. A001906 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2009]
Cf. A178568 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 29 2010]
Cf. A185787, includes a list of sequences based on the natural number array A000027.) [From Clark Kimberling (ck6(AT)evansville.edu) Feb 03 2011]
Sequence in context: A131738 * A001477 A087156 A033619 A130734 A090108
Adjacent sequences: A000024 A000025 A000026 * A000028 A000029 A000030
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KEYWORD
| core,nonn,easy,mult,tabl,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Links edited by Daniel Forgues (squid(AT)zensearch.com), Oct 07 2009
Removed incorrect comment. - Joerg Arndt (arndt(AT)jjj.de), Mar 11 2010
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