cp's OEIS Frontend

A275581 Numbers n such that A010846(n) >= n/2.

%I A275581 #19 Feb 26 2018 03:07:38
%S A275581 1,2,3,4,6,8,10,12,18,30
%N A275581 Numbers n such that A010846(n) >= n/2.
%C A275581 Let 1 <= r <= n be a number "regular to" n, that is, a product of prime divisors p that also divide n.
%C A275581 This sequence includes numbers n such that there are at least as many regulars r among the range 1 <= m <= n as m that are nonregular to n.
%C A275581 Divisors d are a special case of regular r such that d also divides n in addition to all prime divisors p of d also dividing n.
%C A275581 With the exception of r = 1, all regular numbers r are part of the cototient of n.
%C A275581 For prime n, A010846(n) = A000005(n). Composite n > 4 can have nondivisor r. The only prime numbers in a(n) are {2,3}.
%C A275581 Regular r divides some power of n, but reliably divides n^r. When r is a divisor d of n, r | n.
%C A275581 Because the divisors of n are a subset of the regulars of n, it is interesting to compare A010846(n) with A000005(n) = tau(n):
%C A275581 A010846(n) = n for n = (1,2). Tau(n) = n for n = (1,2).
%C A275581 n/2 < A010846(n) < n for n = (3,4,6,10,12,18,30). n/2 < tau(n) < n for n = (3,4,6).
%C A275581 A010846(n) = n/2 for n = 8. Tau(n) = n/2 for n = (8,12).
%C A275581 Sequence A020490 includes numbers n such that tau(n) >= phi(n). The number 24 is in A020490 but not in a(n). This is because A010846(24) = 11 (cf. A162306(24) = {1,2,3,4,6,8,9,12,16,18,24}).
%C A275581 Numbers n such that tau(n) >= n/2: {1,2,3,4,6,8,12}.
%t A275581 Select[Range[10^3], 2 Count[Range@ #, k_ /; PowerMod[#, k, k] == 0] >= # &] (* or *)
%t A275581 Select[Range[10^3], Function[n, 2 Total[MoebiusMu[#] Floor[n/#] &@ Select[Range@ n, CoprimeQ[#, n] &]] >= n]] (* or *)
%t A275581 Select[Range[10^3], Function[n, 2 (1 + Count[Range@ n, m_ /; SubsetQ[FactorInteger[n][[All, 1]], FactorInteger[m][[All, 1]]]]) >= n]]
%Y A275581 Cf. A010846, A020490, A051293, A162306.
%K A275581 nonn,easy,fini,full
%O A275581 1,2
%A A275581 _Michael De Vlieger_, Dec 25 2016