A386572 Numbers that are not of the form k - Omega(k), where Omega(k) is the number of prime factors of k counted with multiplicity.

3, 11, 14, 21, 26, 29, 34, 35, 38, 45, 48, 51, 54, 57, 59, 61, 62, 64, 68, 69, 71, 74, 76, 79, 81, 87, 94, 97, 98, 101, 105, 110, 118, 123, 124, 125, 129, 133, 134, 137, 142, 147, 149, 155, 158, 160, 165, 170, 173, 174, 177, 182, 184, 186, 188, 189, 191, 193, 197

Offset

1,1

Comments

Luca (2005) proved that this sequence is infinite, and Kátai (2006) proved that it has a positive lower density.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Imre Kátai, A remark on a paper of Luca, Acta Mathematica Hungarica, Vol. 113, No. 4 (2006), pp. 313-318.

Florian Luca, On numbers not of the form n-omega(n), Acta Mathematica Hungarica, Vol. 106, No. 1 (2005), pp. 117-135.

Mathematica

seq[lim_] := Complement[Range[lim], Table[k - PrimeOmega[k], {k, 1, lim + Log2[lim]}]]; seq[200]

Prog

(PARI) list(lim) = setminus(vector(lim, i, i), Set(vector(lim + logint(lim, 2), i, i - bigomega(i))));

Crossrefs

Cf. A001222, A069345.

Numbers not of the form k-f(k): A005278 (phi), A045765 (d), A386571 (omega), this sequence (Omega).

Sequence in context: A373955 A143029 A186701 * A022123 A303035 A273869

Adjacent sequences: A386569 A386570 A386571 * A386573 A386574 A386575

Keyword

nonn

Author

Amiram Eldar, Jul 26 2025

Status

approved