A101340 -id:A101340 - cp's OEIS frontend

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

Nathan J. McDougall, Aug 27 2020

nonn

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.

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

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

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).

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

upto(limit)=Set(select(t->t<=limit, apply(m->m+bigomega(m), [1..limit]))) \\ Andrew Howroyd, Aug 27 2020

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

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

A345265 a(n) = Sum_{d|n} n^rad(d).

Original entry on oeis.org

1, 6, 30, 36, 3130, 46914, 823550, 200, 1467, 10000100110, 285311670622, 5973996, 302875106592266, 11112006930971730, 437893890381622140, 1040, 827240261886336764194, 68036454, 1978419655660313589123998, 20480003200820, 5842587018385982523182222244, 341427877364220141714948135418

Offset: 1

Wesley Ivan Hurt, Jun 12 2021

nonn

			a(8) = Sum_{d|8} 8^rad(d) = 8^1 + 8^2 + 8^2 + 8^2 = 200.

Cf. A007947 (rad), A101340.

Table[Sum[(1 - Ceiling[n/i] + Floor[n/i]) n^Product[k^((PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[i/k] + Floor[i/k])), {k, i}], {i, n}], {n, 30}]

rad(n) = factorback(factorint(n)[, 1]);
a(n) = sumdiv(n, d, n^rad(d)); \\ Michel Marcus, Jun 12 2021

a(p) = Sum_{d|p} p^rad(d) = p^1 + p^p = p^p + p, for p prime.

cp's OEIS Frontend

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

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A345265 a(n) = Sum_{d|n} n^rad(d).

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