A103826 -id:A103826 - cp's OEIS frontend

A103827 Arithmetic means of the divisors of unitary arithmetic numbers (i.e., of those for which the arithmetic mean of the unitary divisors is an integer, A103826).

1, 2, 3, 3, 4, 5, 6, 5, 7, 6, 6, 9, 10, 8, 9, 12, 9, 13, 14, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, 15, 15, 18, 24, 17, 25, 18, 27, 21, 18, 18, 20, 30, 15, 31, 24, 20, 21, 18, 34, 24, 18, 36, 37, 26, 25, 24, 21, 40, 41, 42, 20, 27, 33, 30, 27, 45, 28, 30, 32, 36, 30, 33, 49

Offset: 1

Emeric Deutsch, Feb 17 2005

nonn

The unitary arithmetic numbers are in A103826.

			a(8) = 5 because the eighth unitary arithmetic number is A103826(8) = 12, the unitary divisors of 12 are 1, 3, 4 and 12 and (1 + 3 + 4 + 12)/4 = 5.

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

Cf. A034444, A034448, 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 usigma(n)/utau(n) else fi end:seq(p(n),n=1..109);

Select[Table[Mean[Select[Divisors[n],GCD[#,n/#]==1&]],{n,150}],IntegerQ] (* Harvey P. Dale, May 20 2012 *)
Select[Times @@ ((1 + Power @@@ FactorInteger[#])/2) & /@ Range[100], IntegerQ] (* Amiram Eldar, Jun 14 2022 *)

a(n) = A034448(A103826(n))/A034444(A103826(n)). - Amiram Eldar, Jun 19 2019

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]] &]

A353038 Unitary harmonic numbers (A006086) that are not unitary arithmetic numbers (A103826).

Original entry on oeis.org

90, 40682250, 81364500, 105773850, 423095400, 1798155450, 14385243600

Offset: 1

Amiram Eldar, Apr 19 2022

There are 290 unitary harmonic numbers below 10^12, and only 7 of them are in this sequence.

			90 is in the sequence since its unitary divisors are {1, 2, 5, 9, 10, 18, 45, 90}, their harmonic mean, 4, is an integer, but their arithmetic mean, 45/2, is not.

The unitary version of A046999.
Subsequence of A006086.
Cf. A006087, A103826.

q[n_] := Module[{f = FactorInteger[n], d, s}, d = 2^Length[f]; s = Times @@ (1 + Power @@@ f); IntegerQ[n*d/s] && !IntegerQ[s/d]]; Select[Range[5*10^7], q]

A285510 Numbers k such that the average of the squarefree divisors of k is an integer.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 36, 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, 72, 73, 75, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101

Offset: 1

Ilya Gutkovskiy, Apr 20 2017

nonn

Numbers n such that A034444(n)|A048250(n).
Numbers n such that 2^omega(n)|psi(rad(n)), where omega() is the number of distinct prime divisors (A001221), psi() is the Dedekind psi function (A001615) and rad() is the squarefree kernel (A007947).
From Robert Israel, Apr 24 2017: (Start)
All odd numbers are in the sequence.
A positive even number is in the sequence if and only if at least one of its prime factors is in A002145.
Thus this is the complement of 2*A072437 in the positive numbers.
(End)

			44 is in the sequence because 44 has 6 divisors {1, 2, 4, 11, 22, 44} among which 4 are squarefree {1, 2, 11, 22} and (1 + 2 + 11 + 22)/4 = 9 is an integer.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Cf. A001221, A001615, A002145, A003601, A005117, A007947, A023886, A034444, A048250, A072437, A078174, A103826, A206778.

filter:= n -> n::odd or has(numtheory:-factorset(n) mod 4, 3):
select(filter, [$1..1000]); # Robert Israel, Apr 24 2017

Select[Range[100], IntegerQ[Total[Select[Divisors[#], SquareFreeQ]] / 2^PrimeNu[#]] &]
Select[Range[110],IntegerQ[Mean[Select[Divisors[#],SquareFreeQ]]]&] (* Harvey P. Dale, Apr 11 2018 *)
Select[Range[100], IntegerQ[Times @@ ((1 + FactorInteger[#][[;; , 1]])/2)] &] (* Amiram Eldar, Jul 01 2022 *)

a(n) ~ n (conjecture).

Conjecture is true, since A072437 has density 0. - Robert Israel, Apr 24 2017

A353039 Unitary arithmetic numbers k whose mean unitary divisor is a unitary divisor of k.

Original entry on oeis.org

1, 6, 60, 420, 630, 5460, 8190, 16632, 64260, 143640, 172900, 598500, 716625, 790398, 791700, 1182384, 1187550, 1530144, 2708160, 4277448, 5314680, 6284250, 6397300, 6741630, 14619150, 15214500, 22144500, 24315984, 87966648, 93284100, 161670600, 165197760, 232517250

Offset: 1

Amiram Eldar, Apr 19 2022

nonn

Also, unitary harmonic numbers k whose harmonic mean of the unitary divisors of k is a unitary divisor of k.

			6 is a term since the arithmetic mean of its unitary divisors, {1, 2, 3, 6}, is 3, and 3 is also a unitary divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..84

Subsequence of A006086 and A103826.

Cf. A007340.

q[n_] := Module[{f = FactorInteger[n], d, s, m}, d = 2^Length[f]; s = Times @@ (1 + Power @@@ f); m = s/d; IntegerQ[m] && Divisible[n, m] && CoprimeQ[m, n/m]]; Select[Range[10^6], q]

A361386 Infinitary arithmetic numbers: numbers for which the arithmetic mean of the infinitary divisors is an integer.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 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, 89, 91

Offset: 1

Amiram Eldar, Mar 10 2023

nonn

Number k such that A037445(k) divides A049417(k).

Subsequence of the unitary arithmetic numbers (A103826).

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

Cf. A037445, A049417, A077609.

Similar sequences: A003601, A103826.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, (1 + p^(2^(m - j)))/2, 1], {j, 1, m}]]; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], q]

is(n) = {my(f = factor(n), b); denominator(prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], (f[i, 1]^(2^(#b-k))+1)/2, 1)))) == 1; }

6 is a term since the arithmetic mean of its infinitary divisors, {1, 2, 3, 6}, is 3 which is an integer.

A361786 Bi-unitary arithmetic numbers: numbers for which the arithmetic mean of the bi-unitary divisors is an integer.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 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, 89, 91, 92, 93, 94, 95, 96, 97, 99

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

First differs from A361386 at n = 35.

Number k such that A286324(k) divides A188999(k).

			6 is a term since the arithmetic mean of its bi-unitary divisors, {1, 2, 3, 6}, is 3 which is an integer.

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

Cf. A188999, A222266, A286324.

Similar sequences: A003601, A103826, A361386.

f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e]; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], q]

is(n) = {my(f = factor(n), p, e); denominator(prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e))) == 1; }

A192577 Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number.

Original entry on oeis.org

3, 5, 6, 9, 12, 13, 25, 37, 48, 61, 73, 81, 121, 157, 193, 277, 313, 361, 397, 421, 457, 541, 613, 625, 661, 673, 733, 757, 768, 841, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2401, 2473

Offset: 1

Antonio Roldán, Jul 04 2011

nonn

Subsequence of A103826.
Similar to A187073, but considering unitary divisors, not prime divisors.
The odd terms of the sequence are: (1) the terms of A005383 (numbers n such that both n and (n+1)/2 are primes) and (2) the terms of A192618 (prime powers p^k with even exponents k>0 such that (1+p^k)/2 is prime).
[Note that A034448(n) and A034444(n) are multiplicative, so the arithmetic mean A034448(n)/A034444(n) is multiplicative with a(p^e) = (1+p^e)/2.]
The even terms of the sequence are 6, 12, 48, 768, 196608,... (no others < 10^10) with formula n = 3*2^(2^(k-1)) and averages 3, 5, 17, 257, 65537, ... (Fermat numbers, A000215).

			48 has unitary divisors 1, 3, 16, 48 and (1+3+16+48)/4 = 17 is prime, therefore 48 is in the sequence.

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
A. Roldan Martinez, Numeros y hoja de calculo

Cf. A103826, A187073, A005383, A192618, A056798, A000215.

UnitaryDivisors:=func< n | [ d: d in Divisors(n) | Gcd(d, n div d) eq 1 ] >; [ n: n in [1..2500] | IsPrime(k) and s mod #U eq 0 where k is s div #U where s is &+U where U is UnitaryDivisors(n) ]; // Klaus Brockhaus, Jul 09 2011

usigma(n)= {local(f, u=1); f=factor(n); for(i=1, matsize(f)[1], u*=(1+ f[i, 1]^f[i, 2])); return(u)}
ud(n)= {local (f, u); f=factor(n); u=2^(matsize(f)[1]); return(u) }
{  for (n=2, 10^4, c=usigma(n)/ud(n); if (c==truncate(c),if(isprime(c), print1(n, ", ")))) }
\\ Antonio Roldán, Oct 08 2012

A349222 Numbers k such that k and k+1 have the same average of unitary divisors.

Original entry on oeis.org

5, 14, 44, 55, 152, 1334, 1634, 1652, 2204, 2232, 2295, 2685, 3195, 4256, 7191, 8216, 9144, 9503, 9844, 10152, 18423, 19491, 20118, 27404, 30247, 33998, 38180, 42818, 45716, 48364, 51624, 79316, 79338, 84134, 117116, 122073, 124676, 125811, 139460, 157640, 166624

Offset: 1

Amiram Eldar, Nov 11 2021

nonn

The average of the unitary divisors of k is equal to A034448(k)/A034444(k).
Terms k such that k and k+1 are squarefree are also terms of A238380. The terms that are not in A238380 are 44, 55, 152, 1652, 2204, 2232, 2295, 3195, 4256, ...
The average is an integer for the first 1000 terms. Are there terms with a noninteger average?

			5 is a term since the average of the unitary divisors of 5 is (1 + 5)/2 = 3, and the average of the unitary divisors of 6 is (1 + 2 + 3 + 6)/4 = 3.
44 is a term since the average of the unitary divisors of 44 is (1 + 4 + 11 + 44)/4 = 15, and the average of the unitary divisors of 45 is (1 + 5 + 9 + 45)/4 = 15.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007674, A034444, A034448, A103826, A238380.

m[1] = 1; m[n_] := (Times @@ (1 + Power @@@ (f = FactorInteger[n])))/2^Length[f]; Select[Range[10^5], m[#] == m[# + 1] &]

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

A103827 Arithmetic means of the divisors of unitary arithmetic numbers (i.e., of those for which the arithmetic mean of the unitary divisors is an integer, A103826).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A353038 Unitary harmonic numbers (A006086) that are not unitary arithmetic numbers (A103826).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A285510 Numbers k such that the average of the squarefree divisors of k is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A353039 Unitary arithmetic numbers k whose mean unitary divisor is a unitary divisor of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A361386 Infinitary arithmetic numbers: numbers for which the arithmetic mean of the infinitary divisors is an integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A361786 Bi-unitary arithmetic numbers: numbers for which the arithmetic mean of the bi-unitary divisors is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A192577 Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number.

Views

Author

Keywords