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%I #47 Dec 07 2022 17:16:50
%S 1,3,4,6,8,11,12,14,15,16,17,18,20,21,23,24,27,28,30,31,32,33,35,36,
%T 37,38,40,41,42,44,45,47,48,51,53,54,55,57,58,59,60,62,64,66,67,68,69,
%U 70,71,72,73,74,76,77,78,79,80,81,84,85,87,88,89,90,92,93
%N Numbers of the form m + bigomega(m) with m a positive integer.
%C If a(n) = m + A001222(m) then (a(n) - m) <= log(a(n))/log(2).
%C It appears that a(n)/n may converge to a constant around ~ 1.49.
%H Petr Kucheriaviy, <a href="https://arxiv.org/abs/2203.12006">On numbers not representable as n + ω(n)</a>, arXiv preprint (2022). arXiv:2203.12006 [math.NT]
%F 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
%e a(7) = 10 + A001222(10) = 10 + 2 = 12
%t m = 100; Select[Union @ Table[n + PrimeOmega[n], {n, 1, m}], # <= m &] (* _Amiram Eldar_, Aug 28 2020 *)
%o (PARI) upto(limit)=Set(select(t->t<=limit, apply(m->m+bigomega(m), [1..limit]))) \\ _Andrew Howroyd_, Aug 27 2020
%o (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
%Y Cf. A001222 (bigomega), A064800, A358973.
%Y 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).
%Y Subsequences include A008864, A101340, and A160649 (excluding the first term).
%K nonn
%O 1,2
%A _Nathan J. McDougall_, Aug 27 2020
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