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A236999
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Odd part of n*(n+3)/2-1 (A034856).
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3
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1, 1, 1, 13, 19, 13, 17, 43, 53, 1, 19, 89, 103, 59, 67, 151, 169, 47, 13, 229, 251, 137, 149, 323, 349, 47, 101, 433, 463, 247, 263, 559, 593, 157, 83, 701, 739, 389, 409, 859, 901, 59, 247, 1033, 1079, 563, 587, 1223, 1273, 331, 43, 1429, 1483
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OFFSET
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1,4
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COMMENTS
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Let n=2^b*k, where k=k(n) is odd.
Then {odd part(A176126(n-1)), n>=3}={odd part(2^b*k+1)*(2^b*k+2)-4)}={odd part(2^(2*b)*k^2+3*2^b*k-2)}. Hence, if b>0, then {odd part(A176126(n-1), n>=3)= {odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1)}.
On the other hand, in this case odd part(a(n))=odd part(2^(b-1)*k*(2^b*k+3)-1)=odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1). It is left to consider the case of odd n. Setting n=2*m-1, m>=1, we easily find that for both expressions the odd part equals odd part(2*m^2+m-2).
The smallest prime divisor of a(n) is more than or equal to 13.
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LINKS
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MATHEMATICA
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Map[#/2^IntegerExponent[#, 2]&[(# (#+3)/2-1)]&, Range[100]] (* Peter J. C. Moses, Feb 02 2014 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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