%I #11 Jan 12 2024 01:25:18
%S 1,5,6,9,11,12,13,14,15,17,22,23,24,25,27,29,30,33,35,37,38,39,41,42,
%T 43,44,45,47,48,49,53,54,59,60,61,62,65,69,73,76,77,78,81,83,85,86,87,
%U 88,89,91,92,95,96,97,99,101,102,105,107,108,109,110,111,113
%N Unitary arithmetic numbers k (A103826) such that usigma(k)/ud(k) is also a unitary arithmetic number, where ud(k) is the number of divisors of k (A034444) and usigma(k) is their sum (A034448).
%C The number of terms not exceeding 10^k for k = 1, 2, ... is 4, 55, 640, 6990, 74405, 778569, 8050432, 82589241, 842606359, 8562275783, ... Apparently, this sequence has an asymptotic density ~0.85.
%C Includes all the primes p such that (p+1)/2 is an odd prime, i.e., A005383 without the first term 3.
%H Amiram Eldar, <a href="/A334815/b334815.txt">Table of n, a(n) for n = 1..10000</a>
%e 5 is a term since usigma(5)/ud(5) = 6/2 = 3 is an integer, and so is usigma(3)/ud(3) = 4/2 = 2.
%t usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); ud[n_] := 2^PrimeNu[n]; rat[n_] := usigma[n]/ud[n]; Select[Range[200], IntegerQ[(r = rat[#])] && IntegerQ[rat[r]] &]
%Y Cf. A034444, A034448, A103826, A103827, A334813.
%K nonn
%O 1,2
%A _Amiram Eldar_, May 12 2020