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).

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

1,1

Comments

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)

Links

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

Example

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.

Maple

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

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; 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]

Crossrefs

Cf. A000217, A177713.

Sequence in context: A325164 A093642 A364566 * A207675 A118236 A230306

Adjacent sequences: A177730 A177731 A177732 * A177734 A177735 A177736

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 12 2010

Status

approved