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

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

Offset: 1

Author

Vladimir Shevelev, Feb 02 2014

Keywords

• nonn
• easy

Comments

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

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}.

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.

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

Mathematica

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

Formula

a(n) = A000265(A034856(n)). - Michel Marcus, Feb 25 2025