A334813 Arithmetic numbers k (A003601) such that sigma(k)/d(k) is also an arithmetic number, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

1, 5, 6, 11, 13, 14, 15, 20, 29, 37, 38, 39, 41, 43, 44, 45, 49, 53, 54, 56, 57, 59, 60, 61, 65, 68, 73, 78, 83, 85, 86, 87, 89, 95, 96, 97, 101, 102, 107, 109, 110, 111, 113, 114, 116, 118, 123, 125, 129, 131, 134, 135, 137, 139, 142, 143, 145, 147, 150, 153

Offset: 1

Author

Amiram Eldar, May 12 2020

Keywords

• nonn

Comments

The number of terms not exceeding 10^k for k = 1, 2, ... is 3, 36, 426, 4744, 50442, 533806, 5585745, 58013810, 599272790, 6162302702, ... Apparently, this sequence has asymptotic density ~0.6.

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

If p is in A240971 then p^2 is a term.

Examples

			5 is a term since sigma(5)/d(5) = 6/2 = 3 is an integer, and so is sigma(3)/d(3) = 4/2 = 2.
		

Links

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

Crossrefs

Cf. A000005, A000203, A003601, A102187.

Programs

Mathematica

rat[n_] := DivisorSigma[1, n]/DivisorSigma[0, n]; Select[Range[200], IntegerQ[(r = rat[#])] && IntegerQ[rat[r]] &]