%I A000012 M0003
%S A000012 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A000012 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A000012 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A000012 The simplest sequence of positive numbers: the all 1's sequence.
%C A000012 Number of ways of writing n as a product of primes.
%C A000012 Number of ways of writing n as a sum of distinct powers of 2.
%C A000012 Continued fraction for golden ratio A001622.
%C A000012 Partial sums of A000007 (characteristic function of 0). - Jeremy Gardiner
(jeremy.gardiner(AT)btinternet.com), Sep 08 2002
%C A000012 An example of an infinite sequence of positive integers whose distinct
pairwise concatenations are all primes! - Don Reble, Apr 17 2005
%C A000012 Binomial transform of A000007; inverse binomial transform of A000079
. Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 07 2005
%C A000012 A063524(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Oct 11 2008]
%C A000012 For n >= 0, let M(n) be the matrix with 1st row = (n n+1) and 2nd row
= (n+1 n+2). Then a(n) = absolute value of det(M(n)). [From Kailasam
Viswanathan Iyer (kvi(AT)nitt.edu), Apr 11 2009]
%C A000012 The partial sums give the natural numbers (A000027). [From Daniel Forgues
(squid(AT)zensearch.com), May 08 2009]
%C A000012 Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04
2009: (Start)
%C A000012 a(n) is also tau_1(n) where tau_2(n) is A000005
%C A000012 a(n) is a completely multiplicative arithmetical function.
%C A000012 a(n) is both square free and a perfect square. See A005117 and A000290.
(End)
%C A000012 Also smallest divisor of n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Sep 07 2009].
%C A000012 a(n) is also the decimal expansion of 10/9 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es),
Sep 18 2009]
%C A000012 a(n) is also the number of complete graphs on n nodes. [From Pablo Chavez
(pchavez(AT)cmu.edu), Sep 15 2009]
%C A000012 Totally multiplicative sequence with a(p) = 1 for prime p. Totally multiplicative
sequence with a(p) = a(p-1) for prime p. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz),
Oct 18 2009]
%C A000012 nth prime minus phi(prime(n)); number of divisors of n-th prime minus
number of perfect partitions of n-th prime; the number of perfect
partitions of n-th prime number; the number of perfect partitions
of n-th non-composite number. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 26 2009]
%C A000012 Contribution from Harlan J. Brothers (harlan(AT)brotherstechnology.com),
Nov 01 2009: (Start)
%C A000012 For all n>0, the sequence of limit values for a(n)=n!Sum[k=n..inf, k/
(k+1)! ]
%C A000012 Also, for all n != 0, a(n)=n^0 (End)
%C A000012 a(n) is also the number of 0-regular graphs on n vertices. [From Jason
Kimberley (Jason.Kimberley(AT)newcastle.edu.au), Nov 07 2009]
%D A000012 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000012 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 A000012 N. J. A. Sloane, <a href="b000012.txt">Table of n, a(n) for n = 0..1000</
a> [Useful when plotting one sequence against another. See Swayne
link.]
%H A000012 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A000012 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 A000012 N. J. A. Sloane, <a href="a12.html">Illustration of initial terms</a>
%H A000012 D. F. Swayne, <a href="plot2.html">Plot pairs of sequences in the OEIS</
a>
%H A000012 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GoldenRatio.html">Link to a section of The World of Mathematics</
a>
%H A000012 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
ChromaticNumber.html">Chromatic Number</a>
%H A000012 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GraphCycle.html">Graph Cycle</a>
%H A000012 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">
Contfrac</a>
%H A000012 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000012 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
a>
%H A000012 <a href="Sindx_Con.html#confC">Index entries for continued fractions
for constants</a>
%H A000012 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A000012 Harlan Brothers, <a href="http://functions.wolfram.com/GammaBetaErf/Factorial/
23/01/0002/">Factorial: Summation (formula 06.01.23.0002)</a>, The
Wolfram Functions Site [From Harlan J. Brothers (harlan(AT)brotherstechnology.com),
Nov 01 2009]
%F A000012 G.f.: 1/(1-x); a(n)=1. E.g.f.: e^x.
%F A000012 G.f.: Product[(1+x^(2^k)),{k,0,Infinity}]. - Zak Seidov (zakseidov(AT)yahoo.com),
Apr 06 2007
%F A000012 Multiplicative with a(p^e) = 1.
%F A000012 Dirichlet generating function: zeta(s). - Franklin T. Adams-Watters,
Sep 11 2005.
%F A000012 Regarded as a square array by antidiagonals, g.f. 1/((1-x)(1-y)), e.g.f.
sum T(n,m) x^n/n! y^m/m! = e^{x+y}, e.g.f. sum T(n,m) x^n y^m/m!
= e^y/(1-x). Regarded as a triangular array, g.f. 1/((1-x)(1-xy)),
e.g.f. sum T(n,m) x^n y^m/m! = e^{xy}/(1-x). - Frank Adams-Watters
(FrankTAW(AT)Netscape.net), Feb 06 2006
%F A000012 Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04
2009: (Start)
%F A000012 a(n)=Sum(d|n,mu(n/d)*tau_2(d))=1, where tau_2(n)=A000005 and mu(n)=A008683
%F A000012 a(n)=|Sum(d|n,mu(d)*tau_2(d))|=1 (End)
%F A000012 a(n)=A000027(n)-A001477(n). - Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 09 2009
%F A000012 a(n)=A002033(A000040(n))=A002033(A008578(n))=A000005(A000040(n))-A002033(n)=A000027(A000040(n))-A000010(A0000\
40(n)). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct
26 2009]
%e A000012 1.618033988749894848204586834... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...))))
[From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 14 2009]
%p A000012 A000012 := n->1;
%p A000012 [ seq(1,i=0..100) ];
%t A000012 a[n_] := 1
%t A000012 Array[1 &, 50] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26
2006
%t A000012 Table[n!Sum[k/(k+1)!,{k,n,\[Infinity]}],{n, 10}] [From Harlan J. Brothers
(harlan(AT)brotherstechnology.com), Nov 01 2009]
%o A000012 (MAGMA) [ 1 : n in [0..100]];
%o A000012 (PARI) a(n)=1
%o A000012 (PARI) { default(realprecision, 1080); phi = (1 + sqrt(5))/2; x=contfrac(phi);
for (n=1, 1001, write("b000012.txt", n-1, " ", x[n])); } [From Harry
J. Smith (hjsmithh(AT)sbcglobal.net), May 14 2009]
%Y A000012 Cf. A000004, A007395, A010701.
%Y A000012 Cf. A000027.
%Y A000012 Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n):
A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548,
(unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n):
A006218, (tau<=)_3(n): A061201, (tau<=)_5(n): A061203, (tau<=)_6(n):
A061204. [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep
04 2009]
%Y A000012 Cf. A000010, A000040, A002033, A008578. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 26 2009]
%Y A000012 Regular graphs A005176 (any degree), A051031 (triangular array), chosen
degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3),
A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7). [From
Jason Kimberley (Jason.Kimberley(AT)newcastle.edu.au), Nov 07 2009]
%Y A000012 Sequence in context: A087960 A164660 A114523 this_sequence A008836 A064179
A106400
%Y A000012 Adjacent sequences: A000009 A000010 A000011 this_sequence A000013 A000014
A000015
%K A000012 core,easy,nonn,mult,cofr,tabl,new
%O A000012 0,1
%A A000012 N. J. A. Sloane (njas(AT)research.att.com).
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