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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).
1
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
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
Sequence in context: A325164 A093642 A364566 * A207675 A118236 A230306
KEYWORD
nonn
AUTHOR
STATUS
approved