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%I #59 Apr 15 2024 13:12:37
%S 1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U 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 Natural numbers excluding 2.
%C From the following 4 disjoint subsets of natural numbers A = {1}, B = {2}, OP = {odd primes}, C = {composites}, 16 sets are derivable: A000027 versus empty set, A002808 vs A008578, A065091 vs A065090, A000040 vs A018252, A006005 vs {{2} with A002808}, {1} vs {A000027 excluding 1}, {2} versus this sequence, {1, 2} versus Union[OP, C].
%C a(n) is the sum of the obvious divisors of n, which are 1 and n.
%C The natural numbers excluding 2 are the order numbers of magic squares. Order 2 magic squares do not exist. - _William Walkington_, Mar 12 2016
%C The numbers occurring at least twice in Pascal's triangle (A007318, A003016). - _Rick L. Shepherd_, Jun 05 2016
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Magic_square">Magic square</a>.
%H Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 4.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F G.f.: x*(1+x-x^2)/(1-x)^2. - _Paul Barry_, Aug 05 2004
%F a(n) = A000203(n) - A048050(n).
%F a(n) = n+1 for n>1, a(n) = a(n-1)+1 for n>2. - _Wesley Ivan Hurt_, Mar 13 2016
%F E.g.f.: (x + 1)*(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 05 2016
%F a(n) = n + [n>1], a(n) = 1+n-floor(1/n). - _Alan Michael Gómez Calderón_, May 12 2023
%p printlevel := -1; a := [1]; T := x->LambertW(-x); f := series(((1+T(x)))/(1-T(x)), x, 77); for m from 3 to 77 do a := [op(a), op(2*m, f)] od; print(a); # _Zerinvary Lajos_, Mar 28 2009
%t Join[{1}, Range[3, 100]] (* _Wesley Ivan Hurt_, Mar 13 2016 *)
%o (PARI) a(n)=n+(n>1) \\ _Charles R Greathouse IV_, Sep 01 2015
%o (PARI) x='x+O('x^99); Vec((1+x-x^2)/(1-x)^2) \\ _Altug Alkan_, Mar 26 2016
%o (Magma) &cat[[1],[n : n in [3..100]]]; // _Wesley Ivan Hurt_, Mar 13 2016
%Y Cf. A000027, A000040, A000203, A002808, A006005, A008578, A018252, A048050, A065090, A065091, A097330. A003016, A007318.
%K nonn,easy
%O 1,2
%A _Labos Elemer_, Nov 16 2001
%E Incorrect formula removed by _Charles R Greathouse IV_, Mar 18 2010
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