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

1,2

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)

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A285510 A174415 A356844 * A361386 A361786 A079905

Adjacent sequences: A103823 A103824 A103825 * A103827 A103828 A103829

Keyword

nonn

Author

Emeric Deutsch, Feb 17 2005

Status

approved