A065475 Natural numbers excluding 2.

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, 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

Offset: 1

Author

Labos Elemer, Nov 16 2001

Keywords

• nonn
• easy

Comments

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].

a(n) is the sum of the obvious divisors of n, which are 1 and n.

The natural numbers excluding 2 are the order numbers of magic squares. Order 2 magic squares do not exist. - William Walkington, Mar 12 2016

The numbers occurring at least twice in Pascal's triangle (A007318, A003016). - Rick L. Shepherd, Jun 05 2016

From Enrique Navarrete, Mar 03 2025: (Start)

a(n) is the number of binary strings of length n with at most one 0 and at least one 1. For example, the a(1)=1 string is 1 and the a(2)=3 strings are 01, 10, 11.

a(n) is also the number of ordered set partitions of an n-set into 2 sets such that the first set has at most one element and the second set has at least one element. (End)

Links

• Wikipedia, Magic square.
• Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A000027, A000040, A000203, A002808, A006005, A008578, A018252, A048050, A065090, A065091, A097330. A003016, A007318.

Programs

Magma

&cat[[1],[n : n in [3..100]]]; // Wesley Ivan Hurt, Mar 13 2016

Maple

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

Mathematica

Join[{1}, Range[3, 100]] (* Wesley Ivan Hurt, Mar 13 2016 *)
Drop[Range[100],{2}] (* Harvey P. Dale, Aug 11 2024 *)

PARI

a(n)=n+(n>1) \\ Charles R Greathouse IV, Sep 01 2015

PARI

x='x+O('x^99); Vec((1+x-x^2)/(1-x)^2) \\ Altug Alkan, Mar 26 2016

Formula

G.f.: x*(1+x-x^2)/(1-x)^2. - Paul Barry, Aug 05 2004

a(n) = A000203(n) - A048050(n).

a(n) = n+1 for n>1, a(n) = a(n-1)+1 for n>2. - Wesley Ivan Hurt, Mar 13 2016

E.g.f.: (x + 1)*(exp(x) - 1). - Ilya Gutkovskiy, Jun 05 2016

a(n) = n + [n>1], a(n) = 1+n-floor(1/n). - Alan Michael Gómez Calderón, May 12 2023

Extensions

Incorrect formula removed by Charles R Greathouse IV, Mar 18 2010