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

1,1

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

Links

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

Formula

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

Example

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.

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

Prog

(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<<m)-m)<<1 # Chai Wah Wu, Jun 30 2024

Crossrefs

Cf. A000217, A054741.

Intersection of A057716 and A299174.

Related to A051144 via A225546.

Sequence in context: A069209 A060702 A054741 * A176552 A176587 A174164

Adjacent sequences: A177709 A177710 A177711 * A177713 A177714 A177715

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, May 11 2010

Extensions

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

New name from Peter Munn, Jul 31 2020

Status

approved