A177712 Even numbers that have a nontrivial odd divisor.

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, May 11 2010

Keywords

• nonn
• easy

Comments

Numbers which can be expressed as a sum of a set of positive consecutive even numbers: sum_{i=m..m+k} A005843(i), m>=1, k>=1.

Differs from A054741, which contains 105 for example.

These are the numbers that are not free of odd prime factors, but are not odd. Compare with A051144, nonsquarefree nonsquares. The self-inverse function defined by A225546 maps the members of either set 1:1 onto the other set. - Peter Munn, Jul 31 2020 with edit Feb 14 2022

Examples

			6=2+4. 10=4+6. 12=2+4+6. 14=6+8. 18=4+6+8. 20=2+4+6+8. 22=10+12. 24=6+8+10.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000217, A054741.

Intersection of A057716 and A299174.

Related to A051144 via A225546.

Programs

Mathematica

z=200;lst={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst,c]],{b,a-2,1,-2}], {a,2,z,2}];Union@lst

PARI

isA177712(n) = (!(n%2)&&(0<#select(x -> x%2,factor(n)[,1]))); \\ Antti Karttunen, Jul 31 2020

PARI

isA177712(n) = (!(n%2)&&bitand(n,n-1)); \\ Antti Karttunen, Jul 31 2020

Python

def A177712(n): return n+(m:=n.bit_length())+(n>=(1<Chai Wah Wu, Jun 30 2024

Formula

a(n) = 2 * A057716(n).

Extensions

Definition moved into a comment by R. J. Mathar, Aug 15 2010

New name from Peter Munn, Jul 31 2020