A103826 Unitary arithmetic numbers (those for which the arithmetic mean of the unitary divisors is an integer).

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Author

Emeric Deutsch, Feb 17 2005

Keywords

• nonn

Comments

The arithmetic means of the unitary arithmetic numbers are in A103827.

From Amiram Eldar, Mar 10 2023: (Start)

Union of the odd numbers (A005408) and twice the numbers that are not the sum of 2 squares (A022544).

The asymptotic density of this sequence is 1. (End)

Examples

			12 is a unitary arithmetic number because the unitary divisors of 12 are 1,3,4 and 12 and (1+3+4+12)/4=5 is an integer.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005408, A022544, A103827, A034444, A034448.

Programs

Maple

with(numtheory):unitdiv:=proc(n) local A, k: A:={}: for k from 1 to tau(n) do if gcd(divisors(n)[k], n/divisors(n)[k])=1 then A:=A union {divisors(n)[k]} else A:=A fi od end:utau:=n->nops(unitdiv(n)):usigma:=n->add(unitdiv(n)[j],j=1..nops(unitdiv(n))): p:=proc(n) if type(usigma(n)/utau(n),integer)=true then n else fi end:seq(p(n),n=1..103);

Mathematica

udiQ[n_]:=IntegerQ[Mean[Select[Divisors[n],GCD[#,n/#]==1&]]]; Select[ Range[ 100],udiQ] (* Harvey P. Dale, May 20 2012 *)
Select[Range[100], IntegerQ[Times @@ ((1 + Power @@@ FactorInteger[#])/2)] &] (* Amiram Eldar, Jun 14 2022 *)

PARI

is(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^f[i,2]+1)%2^#f~==0 \\ Charles R Greathouse IV, Sep 01 2015