cp's OEIS Frontend

A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

%I A177731 #23 Aug 21 2019 10:57:50
%S A177731 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 A177731 42,44,45,46,48,49,51,53,54,55,56,57,60,61,62,63,65,66,69,70,72,73,75,
%U A177731 76,77,78,81,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,101,102
%N A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.
%C A177731 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 A177731 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 A177731 Does this sequence have asymptotic density 1? - _Robert Israel_, Nov 27 2018
%H A177731 Robert Israel, <a href="/A177731/b177731.txt">Table of n, a(n) for n = 1..10000</a>
%e A177731 5=2+3, 6=1+2+3, 9=4+5, 12=3+4+5,...
%p A177731 f:= proc(n) local r,k;
%p A177731   for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do
%p A177731     k:= (r + 2*n/r - 3)/4;
%p A177731     if k::posint and r >= 2*k+2 then return true fi
%p A177731   od:
%p A177731   false
%p A177731 end proc:
%p A177731 select(f, [$1..1000]); # _Robert Israel_, Nov 27 2018
%t A177731 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 A177731 Cf. A177712, A136724, A177713.
%Y A177731 Cf. A000384, A014106, A071355, A130861, A139577, A139579, A141419.
%Y A177731 Contains A004766, A017137 and nonzero terms of A008588.
%Y A177731 Disjoint from A002145.
%Y A177731 Subsequence of A138591.
%K A177731 nonn
%O A177731 1,1
%A A177731 _Vladimir Joseph Stephan Orlovsky_, May 12 2010