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A294277
Numbers k such that omega(k) < omega(k+1) (where omega(m) = A001221(m), the number of distinct primes dividing m).
10
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
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
KEYWORD
nonn,easy
AUTHOR
Rémy Sigrist, Oct 26 2017
STATUS
approved