login
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).
3
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, May 12 2020
STATUS
approved