A103827 -id:A103827 - cp's OEIS frontend

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

Emeric Deutsch, Feb 17 2005

nonn

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)

			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.

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

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

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);

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 *)

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

A334815 Unitary arithmetic numbers k (A103826) such that usigma(k)/ud(k) is also a unitary arithmetic number, where ud(k) is the number of divisors of k (A034444) and usigma(k) is their sum (A034448).

Original entry on oeis.org

1, 5, 6, 9, 11, 12, 13, 14, 15, 17, 22, 23, 24, 25, 27, 29, 30, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 53, 54, 59, 60, 61, 62, 65, 69, 73, 76, 77, 78, 81, 83, 85, 86, 87, 88, 89, 91, 92, 95, 96, 97, 99, 101, 102, 105, 107, 108, 109, 110, 111, 113

Offset: 1

Amiram Eldar, May 12 2020

nonn

The number of terms not exceeding 10^k for k = 1, 2, ... is 4, 55, 640, 6990, 74405, 778569, 8050432, 82589241, 842606359, 8562275783, ... Apparently, this sequence has an asymptotic density ~0.85.

Includes all the primes p such that (p+1)/2 is an odd prime, i.e., A005383 without the first term 3.

			5 is a term since usigma(5)/ud(5) = 6/2 = 3 is an integer, and so is usigma(3)/ud(3) = 4/2 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034444, A034448, A103826, A103827, A334813.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); ud[n_] := 2^PrimeNu[n]; rat[n_] := usigma[n]/ud[n]; Select[Range[200], IntegerQ[(r = rat[#])] && IntegerQ[rat[r]] &]

A334816 Least number that reaches 1 after n iterations of the map k -> usigma(k)/ud(k) if ud(k) | usigma(k), and k -> 1 otherwise, where ud(k) is the number of unitary divisors of k (A034444) and usigma(k) is their sum (A034448).

Original entry on oeis.org

1, 2, 3, 5, 9, 17, 43, 137, 281, 673, 2401, 4801, 9601, 170761, 341521, 683041, 5114881, 31846081, 131955841, 1985902081, 7545868801

Offset: 0

Amiram Eldar, May 12 2020

Apparently, all the terms are primes or powers of primes.

a(21) > 10^10, if it exists.

			a(3) = 5 since usigma(5)/ud(5) = 6/2 = 3, usigma(3)/ud(3) = 4/2 = 2, and usigma(2)/ud(2) = 3/2 is not an integer, hence there are 3 iterations: 5 -> 3 -> 2 -> 1, and 5 is the least number with 3 iterations.

Cf. A034444, A034448, A103826, A103827, A330814, A330815.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); ud[n_] := 2^PrimeNu[n]; rat[n_] := If[IntegerQ[r = usigma[n]/ud[n]], r, 1]; f[n_] := Length @ FixedPointList[rat, n] - 1; max = 10; seq = Table[0, {max}]; c = 0; n = 1; While[c < max, i = f[n]; If[i <= max && seq[[i]] == 0, c++; seq[[i]] = n]; n++]; seq

cp's OEIS Frontend

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

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A334815 Unitary arithmetic numbers k (A103826) such that usigma(k)/ud(k) is also a unitary arithmetic number, where ud(k) is the number of divisors of k (A034444) and usigma(k) is their sum (A034448).

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A334816 Least number that reaches 1 after n iterations of the map k -> usigma(k)/ud(k) if ud(k) | usigma(k), and k -> 1 otherwise, where ud(k) is the number of unitary divisors of k (A034444) and usigma(k) is their sum (A034448).

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