A177731 - OEIS

%I #23 Aug 21 2019 10:57:50

%S 5,6,9,12,13,14,15,17,18,21,22,24,25,27,28,29,30,33,35,36,37,38,39,41,

%T 42,44,45,46,48,49,51,53,54,55,56,57,60,61,62,63,65,66,69,70,72,73,75,

%U 76,77,78,81,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,101,102

%N Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

%C Numbers of the form sum_{i=j..2k+1} i where j>=1 and 2k+1>j and k>=1. Numbers of the form (2k+1+j)*(2k+2-j)/2, j>=1, k>=1, 2k+1>j. - R. J. Mathar, Dec 04 2011

%C Subsequences include the A000384 where >=6, the A014106 where >=5, A071355 where >=12, A130861 where >=9, A139577 where >=13, A139579 where >=17 etc. The sequence is the union of all odd-indexed rows of A141419, except its first column and numbers <=3: {5,6}, {9,12,14,15}, {13,18,22,25,27,28}, ... - _R. J. Mathar_, Dec 04 2011

%C Does this sequence have asymptotic density 1? - _Robert Israel_, Nov 27 2018

%H Robert Israel, <a href="/A177731/b177731.txt">Table of n, a(n) for n = 1..10000</a>

%e 5=2+3, 6=1+2+3, 9=4+5, 12=3+4+5,...

%p f:= proc(n) local r,k;

%p   for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do

%p     k:= (r + 2*n/r - 3)/4;

%p     if k::posint and r >= 2*k+2 then return true fi

%p   od:

%p   false

%p end proc:

%p select(f, [$1..1000]); # _Robert Israel_, Nov 27 2018

%t z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1

%Y Cf. A177712, A136724, A177713.

%Y Cf. A000384, A014106, A071355, A130861, A139577, A139579, A141419.

%Y Contains A004766, A017137 and nonzero terms of A008588.

%Y Disjoint from A002145.

%Y Subsequence of A138591.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, May 12 2010