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A103826 Unitary arithmetic numbers (those for which the arithmetic mean of the unitary divisors is an integer). 11
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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A285510 A174415 A356844 * A361386 A361786 A079905
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 17 2005
STATUS
approved

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Last modified April 23 18:16 EDT 2024. Contains 371916 sequences. (Running on oeis4.)