A337455 Numbers of the form m + bigomega(m) with m a positive integer.

1, 3, 4, 6, 8, 11, 12, 14, 15, 16, 17, 18, 20, 21, 23, 24, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 51, 53, 54, 55, 57, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 84, 85, 87, 88, 89, 90, 92, 93

Offset

1,2

Comments

If a(n) = m + A001222(m) then (a(n) - m) <= log(a(n))/log(2).

It appears that a(n)/n may converge to a constant around ~ 1.49.

Links

Table of n, a(n) for n=1..66.

Petr Kucheriaviy, On numbers not representable as n + ω(n), arXiv preprint (2022). arXiv:2203.12006 [math.NT]

Formula

Kucheriaviy proves that a(n) << n log log n and conjectures that a(n) ≍ n, that is, these numbers have positive lower density. - Charles R Greathouse IV, Dec 07 2022

Example

a(7) = 10 + A001222(10) = 10 + 2 = 12

Mathematica

m = 100; Select[Union @ Table[n + PrimeOmega[n], {n, 1, m}], # <= m &] (* Amiram Eldar, Aug 28 2020 *)

Prog

(PARI) upto(limit)=Set(select(t->t<=limit, apply(m->m+bigomega(m), [1..limit]))) \\ Andrew Howroyd, Aug 27 2020
(PARI) list(lim)=my(v=List()); forfactored(n=1, lim\1-1, my(t=n[1]+bigomega(n)); if(t<=lim, listput(v, t))); Set(v) \\ Charles R Greathouse IV, Dec 07 2022

Crossrefs

Cf. A001222 (bigomega), A064800, A358973.

Numbers of the form k^n+n where k is prime are subsequences: A006127 (k=2), A104743 (k=3), A104745 (k=5), A226199 (k=7), A226737 (k=11).

Subsequences include A008864, A101340, and A160649 (excluding the first term).

Sequence in context: A277319 A176986 A325455 * A080728 A047414 A109402

Adjacent sequences: A337452 A337453 A337454 * A337456 A337457 A337458

Keyword

nonn

Author

Nathan J. McDougall, Aug 27 2020

Status

approved