A344274 -id:A344274 - cp's OEIS frontend

A344272 a(n) is the least k such that the average number of nonunitary divisors of {1..k} is >= n.

54, 816, 10530, 135200, 1733760, 22216752, 284685408, 3647978320, 46745561100, 599002268832, 7675674748560

Offset: 1

Amiram Eldar, May 13 2021

			a(1) = 54 since the average of the number of nonunitary divisors of {1..54} is (Sum_{k=1..54} A056175(k))/54 = 1.

Cf. A013661, A056175.
The nonunitary version of A085829.
Similar sequences: A328331, A336304, A338891, A338943, A344273, A344274.

nd[n_] := DivisorSigma[0,n] - 2^PrimeNu[n]; seq={}; s = 0; k = 1; Do[While[s = s + nd[k]; s < k*n, k++]; AppendTo[seq, k]; k++, {n, 1, 5}]; seq

Lim_{n->oo} a(n+1)/a(n) = exp(1/(1-1/zeta(2))) = exp(Pi^2/(Pi^2-6)) = 12.8140996101...

a(10)-a(11) from Martin Ehrenstein, May 23 2021

A344273 a(n) is the least k such that the average number of bi-unitary divisors of {1..k} is >= n.

Original entry on oeis.org

1, 6, 24, 80, 273, 960, 3336, 11480, 39648, 136952, 472416, 1630164, 5625480, 19412736, 66992016, 231184800, 797806152, 2753187210, 9501109380, 32787848746

Offset: 1

Amiram Eldar, May 13 2021

			a(2) = 6 since the average number of bi-unitary divisors of {1..6} is A306069(6)/6 = 13/6 > 2.

Cf. A286324, A306069, A306071.
The unitary version of A085829.
Similar sequences: A328331, A336304, A338891, A338943, A344272, A344274.

f[p_, e_] := If[OddQ[e], e + 1, e]; bdivnum[1] = 1; bdivnum[n_] := Times @@ (f @@@ FactorInteger[n]); seq={}; s = 0; k = 1; Do[While[s = s + bdivnum[k]; s < k*n, k++]; AppendTo[seq, k]; k++, {n, 1, 10}]; seq

Lim_{n->oo} a(n+1)/a(n) = exp(1/A) = 3.4509501567..., where A is A306071.

cp's OEIS Frontend

A344272 a(n) is the least k such that the average number of nonunitary divisors of {1..k} is >= n.

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