OFFSET
0,2
COMMENTS
a(0)=1, a(n) = a(n-1) + sigma_0(n) (A000005). - Ctibor O. Zizka, Nov 08 2008
a(n) is the index of the n-th term of A027750 whose value is 1. - Michel Marcus, Oct 15 2015
From Gus Wiseman, Jun 07 2019: (Start)
Also the number of subsets of {1..n} that are closed under taking the difference of two strictly decreasing terms. For example, the a(0) = 1 through a(6) = 15 subsets are:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{1,2} {3} {3} {3} {3}
{1,2} {4} {4} {4}
{1,2,3} {1,2} {5} {5}
{2,4} {1,2} {6}
{1,2,3} {2,4} {1,2}
{1,2,3,4} {1,2,3} {2,4}
{1,2,3,4} {3,6}
{1,2,3,4,5} {1,2,3}
{2,4,6}
{1,2,3,4}
{1,2,3,4,5}
{1,2,3,4,5,6}
(End)
LINKS
Marius A. Burtea, Table of n, a(n) for n = 0..10000 (term 0..1000 from T. D. Noe).
Wikipedia, Arithmetic progression
FORMULA
G.f.: (1-x)^(-1) * (1 + Sum_{j>=1} x^j/(1-x^j)). - Robert Israel, Oct 15 2015
a(n) = Sum_{i=1..n+1} ceiling((n+1)/(i+1)). - Wesley Ivan Hurt, Sep 15 2017
EXAMPLE
a(3)=6 because the six increasing progressions (3), (2,3), (1,2,3), (0,1,2,3), (1,3) and (0,3) all end in 3.
MAPLE
IBI:= {{}}: a[0]:= 1: for n from 1 to 45 do IBI:= IBI union map(t -> t union {n}, select(t -> (t minus map(q -> n-q, t)={}), IBI)); a[n]:= nops(IBI) od: seq(a[n], n=0..45); # Zerinvary Lajos, Mar 18 2007
with(numtheory):a[1]:=2: for n from 2 to 59 do a[n]:=a[n-1]+tau(n) od: seq(a[n], n=0..45); # Zerinvary Lajos, Mar 21 2009
map(`+`, ListTools:-PartialSums(map(numtheory:-tau, [$0..1000])), 1); # Robert Israel, Oct 15 2015
MATHEMATICA
a[0]=1; a[n_] := a[n] = a[n-1] + DivisorSigma[0, n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Oct 05 2012, after Ctibor O. Zizka *)
nxt[{n_, a_}]:={n+1, a+DivisorSigma[0, n+1]}; Transpose[NestList[nxt, {0, 1}, 50]][[2]] (* Harvey P. Dale, Oct 15 2012 *)
Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Subtract@@@Reverse/@Subsets[#, {2}]]&]], {n, 0, 10}] (* Gus Wiseman, Jun 07 2019 *)
PROG
(PARI) vector(100, n, n--; sum(k=1, n, n\k) + 1) \\ Altug Alkan, Oct 15 2015
(Magma) [1] cat [&+[Ceiling((k+1)/(i+1)): i in [1..k+1]]: k in [1..60]]; // Marius A. Burtea, Jun 10 2019
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
Henry Bottomley, Apr 07 2000
STATUS
approved