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A236999 Odd part of n*(n+3)/2-1 (A034856).

%I A236999 #40 Feb 25 2025 04:04:43
%S A236999 1,1,1,13,19,13,17,43,53,1,19,89,103,59,67,151,169,47,13,229,251,137,
%T A236999 149,323,349,47,101,433,463,247,263,559,593,157,83,701,739,389,409,
%U A236999 859,901,59,247,1033,1079,563,587,1223,1273,331,43,1429,1483
%N A236999 Odd part of n*(n+3)/2-1 (A034856).
%C A236999 Also odd part of A176126(n-1) and of |A127276(n-1)|, n>=3.
%C A236999 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 A236999 Let n=2^b*k, where k=k(n) is odd.
%C A236999 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 A236999 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 A236999 The smallest prime divisor of a(n) is more than or equal to 13.
%H A236999 Peter J. C. Moses, <a href="/A236999/b236999.txt">Table of n, a(n) for n = 1..1000</a>
%F A236999 a(n) = A000265(A034856(n)). - _Michel Marcus_, Feb 25 2025
%t A236999 Map[#/2^IntegerExponent[#,2]&[(# (#+3)/2-1)]&,Range[100]] (* _Peter J. C. Moses_, Feb 02 2014 *)
%Y A236999 Cf. A000265, A034856, A176126, A127276, A001788.
%K A236999 nonn,easy
%O A236999 1,4
%A A236999 _Vladimir Shevelev_, Feb 02 2014