A279457 - OEIS

%I #21 Feb 16 2025 08:33:37

%S 2,3,5,7,8,11,13,17,19,23,27,29,30,31,32,37,41,42,43,47,53,59,61,66,

%T 67,70,71,73,78,79,83,89,97,101,102,103,105,107,109,110,113,114,120,

%U 125,127,128,130,131,137,138,139,149,151,154,157,163,165,167,168,170,173,174,179,180,181,182,186,190,191,193,195,197,199,211

%N Numbers k such that number of distinct primes dividing k is odd and number of prime divisors (counted with multiplicity) of k is odd.

%C Intersection of A026424 and A030230.

%C Numbers k such that A000035(A001221(k)) = 1 and A000035(A001222(k)) = 1.

%C Numbers k such that A076479(k) = -1 and A008836(k) = -1.

%C All primes (A000040) are included in the sequence.

%H G. C. Greubel, <a href="/A279457/b279457.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>.

%e 27 is in the sequence because 27 = 3^3 therefore omega(27) = 1 {3} is odd and bigomega(27) = 3 {3,3,3} is odd.

%t Select[Range[220], Mod[PrimeNu[#1], 2] == Mod[PrimeOmega[#1], 2] == 1 & ]

%t Select[Range[300],AllTrue[{PrimeNu[#],PrimeOmega[#]},OddQ]&] (* _Harvey P. Dale_, Jul 10 2023 *)

%o (PARI) is(k) = {my(f = factor(k)); omega(f) % 2 && bigomega(f) % 2;} \\ _Amiram Eldar_, Sep 17 2024

%Y Cf. A000035, A000040, A001221, A001222, A008836, A076479, A026424, A030230, A279456, A279458.

%K nonn,easy,changed

%O 1,1

%A _Ilya Gutkovskiy_, Dec 12 2016