%I A000079 M1129 N0432
%S A000079 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,
%T A000079 262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,67108864,
%U A000079 134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592
%N A000079 Powers of 2: a(n) = 2^n.
%C A000079 Number of subsets of an n-set.
%C A000079 There are 2^(n-1) compositions (ordered partitions) of n - see for example
Riordan. This is the unlabeled analogue of the preferential labelings
sequence A000670.
%C A000079 This is also the number of weakly unimodal permutations of 1..n, that
is, permutations with exactly one local maximum. E.g. a(5)=16: 12345,
12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals.
- Jon Perry (perry(AT)globalnet.co.uk), Jul 27 2003. Proof: see next
line! See also A087783.
%C A000079 Proof: n must appear somewhere and there are 2^(n-1) possible choices
for the subset that precedes it. These must appear in increasing
order and the rest must follow n in decreasing order. QED. - N. J.
A. Sloane (njas(AT)research.att.com), Oct 26, 2003.
%C A000079 a(n+1) = smallest number that is not the sum of any number of (distinct)
earlier terms.
%C A000079 Same as Pisot sequences E(1,2), L(1,2), P(1,2), T(1,2). See A008776 for
definitions of Pisot sequences.
%C A000079 With initial 1 omitted, same as Pisot sequences E(2,4), L(2,4), P(2,4),
T(2,4). - David W. Wilson.
%C A000079 Not the sum of two or more consecutive numbers. - Lekraj Beedassy (blekraj(AT)yahoo.com),
May 14 2004
%C A000079 Least deficient or near-perfect numbers (i.e. n such that sigma(n)=A000203(n)=2n-1).
- Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004. Comment from
Max Alekseyev (maxale(AT)gmail.com), Jan 26 2005: All the powers
of 2 are least deficient numbers but it is not known if there exists
a least deficient number not a power of 2.
%C A000079 The sum of the numbers in the n-th row of Pascal's triangle; the sum
of the coefficients of x in the expansion of (x+1)^n.
%C A000079 The only hailstone sequence which doesn't rebound (except "on the ground").
- Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Jan 29 2005
%C A000079 With p(n) = the number of integer partitions of n, p(i) = the number
of parts of the i-th partition of n, d(i) = the number of different
parts of the i-th partition of n, m(i,j) = multiplicity of the j-th
part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and
prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)}
p(i)!/(prod_{j=1}^{d(i)} m(i,j)!) - Thomas Wieder (wieder.thomas(AT)t-online.de),
May 18 2005
%C A000079 a(n+1) = a(n) XOR 3a(n) where XOR is binary exclusive OR operator. -
Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 19 2005
%C A000079 The number of binary relations on an n-element set that are both symmetric
and antisymmetric. Also the number of binary relations on an n-element
set that are symmetric, antisymmetric and transitive.
%C A000079 An autocopy sequence: its first differences are the sequence itself.
- Alexandre Wajnberg & Eric Angelini (alexandre.wajnberg(AT)ulb.ac.be),
Sep 07 2005
%C A000079 a(n) = largest number with shortest addition chain involving n additions.
- David W. Wilson (davidwwilson(AT)comcast.net), Apr 23 2006
%C A000079 Beginning with a(1) = 0, numbers not equal to the sum of previous distinct
natural numbers. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it),
Aug 06 2006
%C A000079 Smallest order of exactly p(n) nonisomorphic Abelian groups, where p(n)=A000041(n).
{First occurrence of p(n) in A000688(n)} - Lekraj Beedassy (blekraj(AT)yahoo.com),
Jul 11 2006
%C A000079 For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2}
such that for a fixed x in {1,2,...,n} and a fixed y in {1,2]} we
have f(x)<>y. - Aleksandar M. Janjic and Milan R. Janjic (agnus(AT)blic.net),
Mar 27 2007
%C A000079 Let P(A) be the power set of an n-element set A. Then a(n) = the number
of pairs of elements {x,y} of P(A) for which x = y. - Ross La Haye
(rlahaye(AT)new.rr.com), Jan 09 2008 Ross La Haye
%C A000079 a(n)= the number of different ways to run up a staircase with n steps,
taking steps of sizes 1,2,3,... and r (r<=n), where the order IS
important and there is no restriction on the number or the size of
each step taken. - Mohammad K. Azarian (azarian(AT)evansville.edu),
May 21 2008
%C A000079 a(n)=number of permutations on [n+1] such that every initial segment
is an interval of integers. Example: a(3) counts 1234, 2134, 2314,
2341, 3214, 3241, 3421, 4321. The map "p -> ascents of p" is a bijection
from these permutations to subsets of [n]. An ascent of a permutation
p is a position i such that p(i) < p(i+1). The permutations shown
map to 123, 23, 13, 12, 3, 2, 1 and the empty set respectively. -
David Callan (callan(AT)stat.wisc.edu), Jul 25 2008
%C A000079 2^(n-1) is the largest number having n divisors (in the sense of A077569);
A005179(n) is the smallest. [From T. D. Noe (noe(AT)sspectra.com),
Sep 02 2008]
%C A000079 Contribution from Bill R McEachen (bmceachen(AT)centralsan.org), Oct
29 2008: (Start)
%C A000079 a(n) appears to match the number of divisors of the modified primorials
(excluding 2,3and 5)
%C A000079 Very limited range examined, PARI example shown (End)
%C A000079 Successive k such that EulerPhi[k]/k = 1/2. [From Artur Jasinski (grafix(AT)csl.pl),
Nov 07 2008]
%C A000079 A classical transform consists (for general a(n)) in swapping a(2n) and
a(2n+1);examples for Jacobsthal A001045 and successive differences:
A092808,A094359,A140505. a(n)=A000079 leads to 2,1,8,4,32,16,=A135520.
[From Paul Curtz (bpcrtz(AT)free.fr), Jan 05 2009]
%C A000079 This is also the (L)-sieve transform of {2,4,6,8,...,2n,...}=A005843.
(See A152079 for the definition of the (L)-sieve transform.) [From
John W. Layman (layman(AT)math.vt.edu), Jan 23 2009]
%C A000079 a(n) = a(n-1)-th even natural numbers (A005843) for n > 1. [From Jaroslav
Krizek (jaroslav.krizek(AT)atlas.cz), Apr 25 2009]
%C A000079 For n >= 0, a(n) is the number of leaves in a complete binary tree of
height n. For n > 0, a(n) is the number of nodes in an n-cube. [From
Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), May 04 2009]
%C A000079 Permutations of n+1 elements where no element is more than one position
left of its original place. For example, there are 4 such permutations
of three elements: 123, 132, 213, and 312. The 8 such permutations
of four elements are 1234, 1243, 1324, 1423, 2134, 2143, 3124, and
4123. [From Joerg Arndt (arndt(AT)jjj.de), June 24 2009]
%C A000079 Catalan transform of A099087. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Jun 29 2009]
%C A000079 a(n) written in base 2: 1,10,100,1000,10000,..., i.e. (n+1)times 1, n
times 0 (A011557(n)). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz),
Aug 02 2009]
%C A000079 Except for the first term, number n such that if A=(7/8)*n^4; B=(7/16)*n^4;
C=(17/16)*n^4; D=(5/4)*n^4; then A^3+B^3+C^3=D^3 [From Vincenzo Librandi
(vincenzo.librandi(AT)tin.it), Sep 08 2009]
%C A000079 Or, phi(n) is equal to the number of perfect partitions of n. [From Juri-Stepan
Gerasimov (2stepan(AT)rambler.ru), Oct 10 2009]
%C A000079 These are the 2-smooth numbers, positive integers with no prime factors
greater than 2. [From Michael Porter (michael_b_porter(AT)yahoo.com),
Oct 04 2009]
%D A000079 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 1016.
%D A000079 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem,
Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28,
Winter 1997.
%D A000079 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A000079 R. Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969),
456.
%D A000079 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p.
124.
%D A000079 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000079 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000079 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the
Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article
06.1.1.
%H A000079 N. J. A. Sloane, <a href="b000079.txt">Table of n, 2^n for n = 0..1000</
a>
%H A000079 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A000079 Henry Bottomley, <a href="a000079.gif">Illustration of initial terms</
a>
%H A000079 D. Butler, <a href="http://www.tsm-resources.com/alists/pow2.html">Powers
of Two up to 2^222</a>
%H A000079 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000079 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
Publications/books.html">Analytic Combinatorics</a>, 2009; see page
18
%H A000079 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Sequences realized as Parker vectors ...</
a>, J. Integer Seqs., Vol. 6, 2003.
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=6">
Encyclopedia of Combinatorial Structures 6</a>
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=68">
Encyclopedia of Combinatorial Structures 68</a>
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=72">
Encyclopedia of Combinatorial Structures 72</a>
%H A000079 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=267">
Encyclopedia of Combinatorial Structures 267</a>
%H A000079 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas
for Some Functions on Finite Sets</a>
%H A000079 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
The Hankel Transform and Some of its Properties</a>, J. Integer Sequences,
4 (2001), #01.1.5.
%H A000079 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 A000079 G. Villemin's Almanac of Numbers, <a href="http://membres.lycos.fr/villemingerard/
Nombre/Puiss2.htm">Puissances de 2</a>
%H A000079 Sage Weil, <a href="http://www.newdream.net/~sage/old/numbers/pow2.htm">
1058 powers of two</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
FractionalPart.html">Fractional Part</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PowerFractionalParts.html">PowerFractional Parts</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Subset.html">Link to a section of The World of Mathematics (1).</
a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinomialSums.html">Link to a section of The World of Mathematics
(2).</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinomialTransform.html">Link to a section of The World of Mathematics
(3).</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Hypercube.html">Hypercube</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
LeastDeficientNumber.html">Link to a section of The World of Mathematics(4)</
a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CollatzProblem.html">Hailstone Number</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Erf.html">Erf</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Abundance.html">Abundance</a>
%H A000079 Wikipedia, <a href="http://en.wikipedia.org/wiki/Almost_perfect_number">
Almost perfect number</a>
%H A000079 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000079 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A000079 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000079 a(n) = 2^n; a(n) = 2*a(n-1). G.f.: 1/(1-2x), e.g.f.: exp(2x).
%F A000079 2^n = Sum_{k=0..n} binomial(n, k).
%F A000079 a(n) is the number of occurrences of n in A000523. a(n) = A001045(n)
+ A001045(n+1). a(n) = 1 + sum_{k=0..(n-1)} a(k). The Hankel transform
of this sequence gives A000007 = [1, 0, 0, 0, 0, 0, ...]. - DELEHAM
Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004
%F A000079 n such that phi(n)=n/2, for n>1, where phi is the Euler's totient (A000010).
- Lekraj Beedassy (lbeedassy(AT)hotmail.com), Sep 07 2004
%F A000079 This sequence can be generated by the following formula: a(n) = a(n-1)
+ 2*a(n-2) when n > 2; a[1] = 1, a[2] = 2 - Alex Vinokur (alexvn(AT)barak-online.net),
Oct 24 2004
%F A000079 a(n) = StirlingS2(n+1,2) + 1 - Ross La Haye (rlahaye(AT)new.rr.com),
Jan 09 2008 Ross La Haye
%F A000079 This sequence can be generated by a(n+2)=6a(n+1)-8a(n), n=1,2,3,... with
a(1)=1, a(2)=2. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug
06 2008
%F A000079 a(n)=ka(n-1)+(4-2k)a(n-2) for any integer k and n>1, with a(0)=1, a(1)=2.
[From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]
%F A000079 Equals the partition numbers A000041 convolved with A152537. [From Gary
W. Adamson (qntmpkt(AT)yahoo.com), Dec 06 2008]
%F A000079 Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:
%F A000079 a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}
%F A000079 delta(l_1,l_2,...,l_i,...,l_n)
%F A000079 where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i <= l_(i+1) and l_(i+1)
<> 0
%F A000079 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.
%F A000079 G.f.: exp(x)*cosh(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 05 2009]
%F A000079 a(0)=1, a(1)=2; a(n)=a(n-1)^2/a(n-2), n>=2 [From Jaume Oliver Lafont
(joliverlafont(AT)gmail.com), Sep 22 2009]
%F A000079 A000010(a(n))=A002033(a(n)). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 10 2009]
%e A000079 There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1,
2, 3, 12, 13, 23, 123 }.
%e A000079 For n=2, A=14, B=7, C=17, D=20, and 14^3+7^3+17^3=20^3 [From Vincenzo
Librandi (vincenzo.librandi(AT)tin.it), Jun 25 2009]
%p A000079 A000079 := n->2^n; [ seq(2^n,n=0..50) ];
%p A000079 with(combstruct); SeqSetU := [S, {S=Sequence(U), U=Set(Z,card >= 1)},
unlabeled]; seq(count(SeqSetU, size=j),j=1..12);
%p A000079 with(combinat):seq(stirling2(n,2)+1, n=1..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Oct 04 2007
%p A000079 seq(binomial(n+0,0)*2^n,n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jul 08 2008
%p A000079 with(finance):seq(futurevalue(2,1,n), n=-1..31);# [From Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Mar 24 2009]
%p A000079 restart: G(x):=exp(x)*cosh(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],
x) od: x:=0: seq(f[n],n=1..34 );# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 05 2009]
%t A000079 Array[ 2^#&, 50, 0 ]
%t A000079 a = {}; Do[If[EulerPhi[x]/x == 1/2, AppendTo[a, x]], {x, 1, 2048}]; a
[From Artur Jasinski (grafix(AT)csl.pl), Nov 07 2008]
%o A000079 (PARI) a(n)=if(n<0,0,2^n)
%o A000079 (PARI) { unimodal(n)=local(x,d,um,umc); umc=0; for (c=0,n!-1, x=numtoperm(n,
c); d=0; um=1; for (j=2,n,if (x[j]<x[j-1],d=1); if (x[j]>x[j-1] &&
d==1,um=0); if (um==0,break)); if (um==1,print(x)); umc+=um); umc
}
%o A000079 sage: [lucas_number2(n,4,4) for n in xrange(-1,27)] - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jul 08 2008
%o A000079 (PARI) a=7*11*13*17*19*23*29*31*37*41*43*47*53*59*61 %32 = 3909612711980232366109
? b=numdiv(a) %33 = 32768 [From Bill R McEachen (bmceachen(AT)centralsan.org),
Oct 29 2008]
%o A000079 (PARI) { x=1; for (n=0, 1000, write("b000079.txt", n, " ", x); x+=x);
} [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 26 2009]
%Y A000079 a(n) = 2*A001045(n)+A078008(n) = 3*A001045(n)+(-1)^n. - Paul Barry (pbarry(AT)wit.ie),
Feb 20 2003
%Y A000079 Cf. A000225.
%Y A000079 A000079 is the Hankel transform (see A001906 for the definition) of A000984,
A002426, A026375, A026387, A026569, A026585, A026671 and A032351
- John W. Layman (layman(AT)math.vt.edu), Jul 31 2000
%Y A000079 Euler transform of A001037.
%Y A000079 Complement of A057716.
%Y A000079 a(n) = A118654(n, 2).
%Y A000079 a(n) = A140740(n+1, 1).
%Y A000079 Cf. A038754, A133464, A140730, A037124.
%Y A000079 Cf. A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325,
A140354.
%Y A000079 Cf. A000041, A152537 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec
06 2008]
%Y A000079 Equals row sums of the partition convolution triangle, A152538 [From
Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 10 2008]
%Y A000079 Cf. A000010, A002033.[From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 10 2009]
%Y A000079 Sequence in context: A166444 A084633 A122803 this_sequence A120617 A050732
A138815
%Y A000079 Adjacent sequences: A000076 A000077 A000078 this_sequence A000080 A000081
A000082
%K A000079 core,easy,nice,nonn
%O A000079 0,2
%A A000079 N. J. A. Sloane (njas(AT)research.att.com).
%E A000079 Clarified a comment T. D. Noe (noe(AT)sspectra.com), Aug 30 2009
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