cp's OEIS Frontend

A177712 Even numbers that have a nontrivial odd divisor.

%I A177712 #21 Jun 30 2024 22:09:37
%S A177712 6,10,12,14,18,20,22,24,26,28,30,34,36,38,40,42,44,46,48,50,52,54,56,
%T A177712 58,60,62,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,
%U A177712 104,106,108,110,112,114,116,118,120,122,124,126,130,132,134,136,138,140
%N A177712 Even numbers that have a nontrivial odd divisor.
%C A177712 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.
%C A177712 Differs from A054741, which contains 105 for example.
%C A177712 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
%H A177712 Antti Karttunen, <a href="/A177712/b177712.txt">Table of n, a(n) for n = 1..20000</a>
%F A177712 a(n) = 2 * A057716(n).
%e A177712 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.
%t A177712 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
%o A177712 (PARI) isA177712(n) = (!(n%2)&&(0<#select(x -> x%2,factor(n)[,1]))); \\ _Antti Karttunen_, Jul 31 2020
%o A177712 (PARI) isA177712(n) = (!(n%2)&&bitand(n,n-1)); \\ _Antti Karttunen_, Jul 31 2020
%o A177712 (Python)
%o A177712 def A177712(n): return n+(m:=n.bit_length())+(n>=(1<<m)-m)<<1 # _Chai Wah Wu_, Jun 30 2024
%Y A177712 Cf. A000217, A054741.
%Y A177712 Intersection of A057716 and A299174.
%Y A177712 Related to A051144 via A225546.
%K A177712 nonn,easy
%O A177712 1,1
%A A177712 _Vladimir Joseph Stephan Orlovsky_, May 11 2010
%E A177712 Definition moved into a comment by _R. J. Mathar_, Aug 15 2010
%E A177712 New name from _Peter Munn_, Jul 31 2020