A177731 -id:A177731 - cp's OEIS frontend

A177732 The sums of two or more consecutive positive numbers, the largest being even.

3, 7, 9, 10, 11, 15, 18, 19, 20, 21, 23, 26, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 43, 45, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 63, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 87, 90, 91, 93, 95, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form (j+2l)*(2l-j+1)/2 with j>=1 and 2l>j. Subsequences are A014105 where >=3, (j=1), A014107 where >=9 (j=2). - R. J. Mathar, Jul 14 2012

			3=1+2, 7=3+4, 9=2+3+4, 10=1+2+3+4, 11=5+6,..

Cf. A177712, A177713, A177731.

z=200;lst2={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst2,c]],{b,a-1,1,-1}],{a,2,z,2}];Union@lst2
With[{upto=108},Select[Union[Flatten[Table[Accumulate[Range[2n-1,1,-1]]+ 2n,{n,upto/4}]]],#<=upto&]] (* Harvey P. Dale, May 19 2019 *)

A177733 Integers that can be expressed as the sum of two or more positive consecutive numbers (the largest being even) AND also as the sum of two or more positive consecutive numbers (the largest being odd).

Original entry on oeis.org

9, 15, 18, 21, 27, 30, 33, 35, 36, 39, 42, 45, 49, 51, 54, 55, 57, 60, 63, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 91, 93, 95, 98, 99, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 126, 129, 132, 133, 135, 138, 140, 141, 143, 144, 147, 150, 153, 154

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Intersection of A177732 and A177731.
From Robert Israel, May 02 2023: (Start)
Numbers k with odd divisors d_1, d_2 >= 2 such that k + (d_1+1)/2 is odd and
k + (d_2+1)/2 is even.
Contains no primes, powers of 2 or products of a prime and a power of 2.
Contains odd semiprime p*q iff at least one of p and q == 3 (mod 4).
(End)

			9 is in the sequence because 2+3+4=9=4+5.
15 is in the sequence because 7+8=15=1+2+3+4+5.

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

Cf. A000217, A177713.

filter:= proc(n) local a,b,x,y,todd,teven;
   todd:= false; teven:= false;
   for a in select(type,numtheory:-divisors(n),odd) minus {1} do
     b:= 2*n/a;
     x:= (a+b+1)/2;
       if x::odd then todd:= true; if teven then return true fi
       else teven:= true; if todd then return true fi
     fi od:
  false
end proc:
select(filter, [$1..200]); # Robert Israel, May 01 2023

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; z=200;lst2={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst2,c]],{b,a-1,1,-1}],{a,2,z,2}]; Intersection[lst1,lst2]

cp's OEIS Frontend

A177732 The sums of two or more consecutive positive numbers, the largest being even.

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