A279457 Numbers k such that number of distinct primes dividing k is odd and number of prime divisors (counted with multiplicity) of k is odd.

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 30, 31, 32, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 120, 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

Offset: 1

Author

Ilya Gutkovskiy, Dec 12 2016

Keywords

• nonn
• easy

Comments

Intersection of A026424 and A030230.

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

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

All primes (A000040) are included in the sequence.

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Distinct Prime Factors.
• Eric Weisstein's World of Mathematics, Prime Factor.

Crossrefs

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

Programs

Mathematica

Select[Range[220], Mod[PrimeNu[#1], 2] == Mod[PrimeOmega[#1], 2] == 1 & ]
Select[Range[300],AllTrue[{PrimeNu[#],PrimeOmega[#]},OddQ]&] (* Harvey P. Dale, Jul 10 2023 *)

PARI

is(k) = {my(f = factor(k)); omega(f) % 2 && bigomega(f) % 2;} \\ Amiram Eldar, Sep 17 2024