A294277 Numbers k such that omega(k) < omega(k+1) (where omega(m) = A001221(m), the number of distinct primes dividing m).

1, 5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 104, 107, 109, 113, 119, 121, 125, 128, 129, 131, 137, 139, 149, 151, 153, 155, 157, 163, 164, 167, 169, 173, 179, 181, 185

Offset: 1

Author

Rémy Sigrist, Oct 26 2017

Keywords

• nonn
• easy

Comments

This sequence, alongside A006049 and A294278, form a partition of the positive integers.

The asymptotic density of this sequence is 1/2 (Erdős, 1936). - Amiram Eldar, Sep 17 2024

Examples

			omega(1) = 0 < omega(2) = 1, hence 1 belongs to this sequence.
omega(4) = 1 = omega(5) = 1, hence 4 does not belong to this sequence.
omega(6) = 2 > omega(7) = 1, hence 6 does not belong to this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Crossrefs

Cf. A001221, A006049, A294278.

Programs

Mathematica

Position[Partition[PrimeNu[Range[200]],2,1],?(#[[1]]<#[[2]]&),1,Heads-> False]//Flatten (* _Harvey P. Dale, May 06 2018 *)

PARI

for (n=1, 185, if (omega(n) < omega(n+1), print1 (n ", ")))