login
Odd part of n*(n+3)/2-1 (A034856).
3

%I #35 Feb 06 2014 04:57:19

%S 1,1,1,13,19,13,17,43,53,1,19,89,103,59,67,151,169,47,13,229,251,137,

%T 149,323,349,47,101,433,463,247,263,559,593,157,83,701,739,389,409,

%U 859,901,59,247,1033,1079,563,587,1223,1273,331,43,1429,1483

%N Odd part of n*(n+3)/2-1 (A034856).

%C Also odd part of A176126(n-1) and of |A127276(n-1)|, n>=3.

%C Proof. By A127276 and A001788, we have odd part(A176126(n))=odd part(|A127276(n)|) = odd part(n*(n+1)-4), {odd part(A176126(n-1)), n>=3}={odd part((n+1)*(n+2)-4), n>=1}.

%C Let n=2^b*k, where k=k(n) is odd.

%C 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)}.

%C 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).

%C The smallest prime divisor of a(n) is more than or equal to 13.

%H Peter J. C. Moses, <a href="/A236999/b236999.txt">Table of n, a(n) for n = 1..1000</a>

%t Map[#/2^IntegerExponent[#,2]&[(# (#+3)/2-1)]&,Range[100]] (* _Peter J. C. Moses_, Feb 02 2014 *)

%Y Cf. A000265, A034856, A176126, A127276, A001788.

%K nonn,easy

%O 1,4

%A _Vladimir Shevelev_, Feb 02 2014