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A026424 Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative. 197
2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 30, 31, 32, 37, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Neither this sequence nor its complement (A028260) contains any infinite arithmetic progression. - Franklin T. Adams-Watters, Sep 05 2008
A066829(a(n)) = 1. - Reinhard Zumkeller, Jun 26 2009
These numbers can be generated by the sieving process described in A066829. - Reinhard Zumkeller, Jul 01 2009
Lexicographically earliest sequence of distinct nonnegative integers with no term being the product of any two not necessarily distinct terms. The equivalent sequence for addition/subtraction is A005408 (the odd numbers), for exponentiation is A259444, and for binary exclusive OR is A000069. - Peter Munn, Mar 16 2018
The equivalent lexicographically earliest sequence with no term being the product of any two distinct terms is A026416. A000028 is similarly the equivalent sequence when A059897 is used as multiplicative operator in place of standard integer multiplication. - Peter Munn, Mar 16 2019
LINKS
S. Ramanujan, Irregular numbers, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21.
Eric Weisstein's World of Mathematics, Prime Sums
FORMULA
Sum 1/a(n)^m = (zeta(m)^2-zeta(2m))/(2*zeta(m)), Dirichlet g.f. of A066829. - Ramanujan.
n>=2 is in sequence if n is not the product of two smaller elements. - David W. Wilson, May 06 2005
A001222(a(n)) mod 2 = 1. - Reinhard Zumkeller, Oct 05 2011
Union of A000040, A014612, A014614, A046308 etc. - R. J. Mathar, Jul 09 2012
MAPLE
isA026424 := proc(n)
if type(numtheory[bigomega](n) , 'odd') then
true;
else
false;
end if;
end proc:
A026424 := proc(n)
option remember;
if n =1 then
2;
else
for a from procname(n-1)+1 do
if isA026424(a) then
return a;
end if;
end do:
end if;
end proc: # R. J. Mathar, May 25 2017
MATHEMATICA
Select[Range[2, 112], OddQ[Total[FactorInteger[#]][[2]]] &] (* T. D. Noe, May 07 2011 *)
(* From version 7 on *) Select[Range[2, 112], LiouvilleLambda[#] == -1 &] (* Jean-François Alcover, Aug 19 2013 *)
PROG
(Haskell)
a026424 n = a026424_list !! (n-1)
a026424_list = filter (odd . a001222) [1..]
-- Reinhard Zumkeller, Oct 05 2011
(PARI) is(n)=bigomega(n)%2 \\ Charles R Greathouse IV, Sep 16 2015
CROSSREFS
Cf. A008836, A028260 (complement).
Apart from initial term, same as A026422.
Cf. A026416 and cross-references therein.
Sequence in context: A359766 A166982 A026422 * A298207 A347453 A229125
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)