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A177733
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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).
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1
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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
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OFFSET
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1,1
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COMMENTS
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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)
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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