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

1,2

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.

Links

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

Example

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.

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]] &]

Crossrefs

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

Sequence in context: A246784 A134736 A285505 * A220968 A179662 A137859

Adjacent sequences: A334812 A334813 A334814 * A334816 A334817 A334818

Keyword

nonn

Author

Amiram Eldar, May 12 2020

Status

approved