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

5, 6, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 53, 54, 55, 56, 57, 60, 61, 62, 63, 65, 66, 69, 70, 72, 73, 75, 76, 77, 78, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102

Offset

1,1

Comments

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

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

Does this sequence have asymptotic density 1? - Robert Israel, Nov 27 2018

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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

Maple

f:= proc(n) local r, k;
  for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do
    k:= (r + 2*n/r - 3)/4;
    if k::posint and r >= 2*k+2 then return true fi
  od:
  false
end proc:
select(f, [$1..1000]); # Robert Israel, Nov 27 2018

Mathematica

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

Crossrefs

Cf. A177712, A136724, A177713.

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

Contains A004766, A017137 and nonzero terms of A008588.

Disjoint from A002145.

Subsequence of A138591.

Sequence in context: A179662 A137859 A082576 * A074185 A242733 A195928

Adjacent sequences: A177728 A177729 A177730 * A177732 A177733 A177734

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 12 2010

Status

approved