A023886 -id:A023886 - cp's OEIS frontend

A284288 Numbers n such that the average of the strong divisors of n is an integer.

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 28, 29, 31, 37, 41, 43, 47, 49, 53, 54, 56, 59, 61, 64, 67, 68, 71, 73, 79, 81, 83, 89, 91, 97, 98, 99, 100, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 138, 139, 148, 149, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 188, 191, 192, 193, 197, 199

Offset: 1

Ilya Gutkovskiy, Mar 24 2017

nonn

We say d is a strong divisor of n iff d is a divisor of n and d > 1.
Numbers n such that A032741(n) divides A039653(n).
All primes and squares of primes are in this sequence.
Positions of ones in A296082 and A296084. - Antti Karttunen, Dec 05 2017

			28 is in the sequence because 28 has 6 divisors {1, 2, 4, 7, 14, 28} therefore 5 strong divisors {2, 4, 7, 14, 28}, 2 + 4 + 7 + 14 + 28 = 55 and 5 divides 55.

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

Cf. A000203, A000430, A003601, A023884, A023886, A032741, A039653, A296082, A296084 (characteristic function).

filter:= proc(n) local d,t;
  d:= numtheory:-divisors(n) minus {1};
  convert(d,`+`) mod nops(d) = 0
end proc:
select(filter, [$2..1000]); # Robert Israel, Mar 27 2017

Select[Range[2, 200], Mod[DivisorSigma[1, #1] - 1, DivisorSigma[0, #1] - 1] == 0 &]

for(n=2, 200, if((sigma(n) - 1)%(numdiv(n) - 1)==0, print1(n,", "))) \\ Indranil Ghosh, Mar 24 2017

from sympy.ntheory import divisor_sigma, divisor_count
print([n for n in range(2, 201) if (divisor_sigma(n) - 1)%(divisor_count(n) - 1) == 0]) # Indranil Ghosh, Mar 24 2017

A284755 Numbers n such that the average of all proper divisors of all positive integers <= n is an integer.

Original entry on oeis.org

2, 3, 63, 1249, 4696, 1200509

Offset: 1

Ilya Gutkovskiy, Apr 02 2017

Numbers n such that A002541(n)|A153485(n).

a(7) > 10^12. - Giovanni Resta, Apr 13 2017

Eric Weisstein's World of Mathematics, Restricted Divisor Function
Index entries for sequences related to sums of divisors

Cf. A000005, A000203, A001065, A002541, A003601, A023884, A023886, A027751, A153485, A226647, A284288.

Select[Range[2, 1300000], Mod[Sum[DivisorSigma[1, k] - k, {k, 1, #}], Sum[DivisorSigma[0, k] - 1, {k, 1, #}]] == 0 &]

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

A247136 Numbers for which the root mean square of nontrivial divisors is an integer.

Original entry on oeis.org

4, 9, 25, 49, 119, 121, 161, 169, 289, 343, 361, 369, 527, 529, 711, 721, 833, 841, 959, 961, 1081, 1127, 1241, 1369, 1681, 1695, 1767, 1849, 2047, 2209, 2809, 3281, 3335, 3481, 3553, 3713, 3721, 4207, 4489, 4633, 4681, 5041, 5047, 5215, 5329, 6241, 6713, 6887

Offset: 1

Daniel Lignon, Nov 20 2014

nonn

All the squares of prime numbers (A001248) have this property but there are other numbers (A247137): 119,161,343,369,527,711,721,833,959,1081...

			119 is a term because the nontrivial divisors of 119 are [7,17] and sqrt((7^2+17^2)/2)= 13 : it's an integer.

Daniel Lignon and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 316 terms from Lignon)

Cf. A140480 (numbers for which the root mean square of all divisors is an integer), A247136 (numbers for which the root mean square of proper divisors is an integer) and A023886 (numbers for which the arithmetic mean of nontrivial divisors is an integer).

Select[Range[2,100000],(Not[PrimeQ[#]] && IntegerQ[ RootMeanSquare[ Rest[ Most[ Divisors[#]]]]])&]

integralRMS(v)=my(t=norml2(v)/#v); denominator(t)==1 && issquare(t)
is(n)=my(d=divisors(n)); #d>2 && integralRMS(d[2..#d-1]) \\ Charles R Greathouse IV, Nov 20 2014

Trivially a(n) << n^2 log^2 n. - Charles R Greathouse IV, Nov 20 2014

A286972 Numbers k such that the average of the prime power divisors (not including 1) of k is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 42, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 119, 121, 123, 127, 129, 131, 132, 133, 135, 137, 139

Offset: 1

Ilya Gutkovskiy, May 17 2017

nonn

Numbers k such that A001222(k)|A023889(k).

			12 is in the sequence because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are prime powers {2, 3, 4} and (2 + 3 + 4)/3 = 3 is an integer.

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

Cf. A001222, A003601, A023886, A023889, A078174, A246655, A285510.

fQ[n_] := IntegerQ@Mean@Select[Divisors@n, PrimePowerQ]; Select[Range@140, fQ]

isok(m) = my(vd = select(isprimepower, divisors(m))); #vd && !(vecsum(vd) % #vd); \\ Michel Marcus, Apr 28 2020

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A284288 Numbers n such that the average of the strong divisors of n is an integer.

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A284755 Numbers n such that the average of all proper divisors of all positive integers <= n is an integer.

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

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A247136 Numbers for which the root mean square of nontrivial divisors is an integer.

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A286972 Numbers k such that the average of the prime power divisors (not including 1) of k is an integer.

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