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

1,2

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

Cf. A000005, A000203, A003601, A102187.

Sequence in context: A332482 A190895 A129286 * A083450 A136974 A101187

Adjacent sequences: A334810 A334811 A334812 * A334814 A334815 A334816

Keyword

nonn

Author

Amiram Eldar, May 12 2020

Status

approved