cp's OEIS Frontend

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

%I A103826 #19 Dec 10 2024 05:23:21
%S A103826 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,
%T A103826 37,38,39,41,42,43,44,45,46,47,48,49,51,53,54,55,56,57,59,60,61,62,63,
%U A103826 65,66,67,69,70,71,73,75,76,77,78,79,81,83,84,85,86,87,88,89,91,92,93
%N A103826 Unitary arithmetic numbers (those for which the arithmetic mean of the unitary divisors is an integer).
%C A103826 The arithmetic means of the unitary arithmetic numbers are in A103827.
%C A103826 From _Amiram Eldar_, Mar 10 2023: (Start)
%C A103826 Union of the odd numbers (A005408) and twice the numbers that are not the sum of 2 squares (A022544).
%C A103826 The asymptotic density of this sequence is 1. (End)
%H A103826 Charles R Greathouse IV, <a href="/A103826/b103826.txt">Table of n, a(n) for n = 1..10000</a>
%e A103826 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.
%p A103826 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);
%t A103826 udiQ[n_]:=IntegerQ[Mean[Select[Divisors[n],GCD[#,n/#]==1&]]]; Select[ Range[ 100],udiQ] (* _Harvey P. Dale_, May 20 2012 *)
%t A103826 Select[Range[100], IntegerQ[Times @@ ((1 + Power @@@ FactorInteger[#])/2)] &] (* _Amiram Eldar_, Jun 14 2022 *)
%o A103826 (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
%Y A103826 Cf. A005408, A022544, A103827, A034444, A034448.
%K A103826 nonn
%O A103826 1,2
%A A103826 _Emeric Deutsch_, Feb 17 2005