A026424 Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

• T. D. Noe, Table of n, a(n) for n = 1..10000
• S. Ramanujan, Irregular numbers, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21.
• Eric Weisstein's World of Mathematics, Prime Sums
• Index entries for sequences generated by sieves - _Reinhard Zumkeller_, Jul 01 2009

Crossrefs

Cf. A008836, A028260 (complement).

Apart from initial term, same as A026422.

Cf. A026416 and cross-references therein.

Cf. A000028, A000069, A005408, A259444.

Programs

Haskell

a026424 n = a026424_list !! (n-1)
a026424_list = filter (odd . a001222) [1..]
-- Reinhard Zumkeller, Oct 05 2011

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 *)
Select[Range[150],OddQ[PrimeOmega[#]]&] (* Harvey P. Dale, Oct 04 2024 *)

PARI

is(n)=bigomega(n)%2 \\ Charles R Greathouse IV, Sep 16 2015

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A026424(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+1+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,m)) for m in range(2,x.bit_length()+1,2)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

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