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

1,2

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

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A001221, A006049, A294278.

Sequence in context: A314584 A335486 A347771 * A373674 A043721 A043727

Adjacent sequences: A294274 A294275 A294276 * A294278 A294279 A294280

Keyword

nonn,easy,changed

Author

Rémy Sigrist, Oct 26 2017

Status

approved