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A177712
Even numbers that have a nontrivial odd divisor.
6
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
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
Intersection of A057716 and A299174.
Related to A051144 via A225546.
Sequence in context: A069209 A060702 A054741 * A176552 A176587 A174164
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Definition moved into a comment by R. J. Mathar, Aug 15 2010
New name from Peter Munn, Jul 31 2020
STATUS
approved