A334815 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).

1, 5, 6, 9, 11, 12, 13, 14, 15, 17, 22, 23, 24, 25, 27, 29, 30, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 53, 54, 59, 60, 61, 62, 65, 69, 73, 76, 77, 78, 81, 83, 85, 86, 87, 88, 89, 91, 92, 95, 96, 97, 99, 101, 102, 105, 107, 108, 109, 110, 111, 113

Offset: 1

Author

Amiram Eldar, May 12 2020

Keywords

• nonn

Comments

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.

Includes all the primes p such that (p+1)/2 is an odd prime, i.e., A005383 without the first term 3.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034444, A034448, A103826, A103827, A334813.

Programs

Mathematica

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]] &]