A206778 -id:A206778 - cp's OEIS frontend

A050328 Number of ordered factorizations of n into squarefree numbers > 1.

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 3, 3, 1, 1, 5, 1, 5, 3, 3, 1, 7, 1, 3, 1, 5, 1, 13, 1, 1, 3, 3, 3, 13, 1, 3, 3, 7, 1, 13, 1, 5, 5, 3, 1, 9, 1, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 31, 1, 3, 5, 1, 3, 13, 1, 5, 3, 13, 1, 25, 1, 3, 5, 5, 3, 13, 1, 9, 1, 3, 1, 31, 3, 3, 3, 7, 1, 31, 3, 5, 3, 3, 3, 11, 1, 5

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000670, A000961, A002033, A002110, A005117, A008966, A050329, A114006, A206778.

import Data.List (genericIndex)
a050328 n = genericIndex a050328_list (n-1)
a050328_list = f 1 where
   f x = (if x == 1 then 1 else
         sum $ map (a050328 . (div x)) $ tail $ a206778_row x) : f (x + 1)
-- Reinhard Zumkeller, May 03 2013

a[n_]:=If[n==1, n, Sum[If[(dIndranil Ghosh, May 27 2017 *)

A050328(n) = if(1==n,n,sumdiv(n,d,if((dA050328(d),0))); \\ Antti Karttunen, May 27 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of squarefree numbers > 1.
a(A000961(n)) = 1.
a(A002110(n)) = A000670(n).
a(n) = Sum_{d divides n, dA008966(n/d)*a(d). - Vladeta Jovovic, Sep 25 2002, corrected by Antti Karttunen, May 27 2017
G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} mu(k)^2*A(x^k). - Ilya Gutkovskiy, May 10 2019

A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 180, 182, 186, 190, 195, 222, 230, 231, 238, 246, 252, 255, 258, 266, 273, 282, 285, 286, 290, 300, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 396, 399, 402, 406, 410, 418, 426, 429

Offset: 1

Reinhard Zumkeller, May 03 2013

nonn

Union of A007304, A179643 and A179695; subsequence of A033992;
A001221(a(n)) = 3 and A051903(a(n)) <= A051904(a(n)) + 1 and A001222(a(n)) = 3 or 5 or 7;
A050326(a(n)) = 5.

			A007304(1) = 2*3*5 = 30, A206778(30,1..8)=[1,2,3,5,6,10,15,30]:
A050326(30) = #{30, 15*2, 10*3, 6*5, 5*3*2} = 5;
A179643(1) = 2^2*3^2*5 = 180, A206778(180,1..8)=[1,2,3,5,6,10,15,30]:
A050326(180) = #{30*6, 30*3*2, 15*6*2, 10*6*3, 6*5*3*2} = 5;
A179695(1) = 2^3*3^2*5^2 = 1800, A206778(1800,1..8)=[1,2,3,5,6,10,15,30]:
A050326(1800) = #{30*10*6, 30*6*5*2, 30*10*3*2, 15*10*6*2, 10*6*5*3*2} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A124010.

a225228 n = a225228_list !! (n-1)
a225228_list = filter f [1..] where
   f x = length es == 3 && sum es `elem` [3,5,7] &&
                           maximum es - minimum es <= 1
         where es = a124010_row x

is(n)=my(f=vecsort(factor(n)[,2]~)); f==[1,1,1] || f==[1,2,2] || f==[2,2,3] \\ Charles R Greathouse IV, Jul 28 2016

a(n) ~ 2n log n / (log log n)^2. - Charles R Greathouse IV, Jul 28 2016

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

A291109 Numbers that are not the sum of the squarefree divisors of some natural number.

Original entry on oeis.org

2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Impossible values for A048250 (numbers k in increasing order such that A048250(m) = k has no solution).
Numbers that are not of the form Product (p_i + 1), p is a prime, so all odd numbers (except 1 and 3) are in this sequence.
Also numbers that are not the sum of the divisors of some squarefree number.

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

Cf. A005114, A005117, A007369, A048250, A048995, A062822, A063948, A064000, A203444, A206778, A274793.

sort(convert({$1..1000} minus map(numtheory:-sigma, select(numtheory:-issqrfree, {$1..1000})),list)); # Robert Israel, Jun 26 2018

TakeWhile[Complement[Range@ #, Union@ Table[Total@ Select[Divisors@ n, SquareFreeQ], {n, 2 #}]], Function[k, k <= #]] &@ 111

A295908 Triangle in which n-th row lists divisors d of n such that n/d is squarefree.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 4, 8, 3, 9, 1, 2, 5, 10, 1, 11, 2, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 8, 16, 1, 17, 3, 6, 9, 18, 1, 19, 2, 4, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 4, 8, 12, 24, 5, 25, 1, 2, 13, 26, 9, 27, 2, 4, 14, 28

Offset: 1

Rémy Sigrist, Nov 29 2017

For any n > 0:
- the n-th row has A034444(n) terms,
- the n-th row has sum A001615(n),
- the n-th row has leading term A003557(n).

			Triangle begins:
  1:  [1]
  2:  [1, 2]
  3:  [1, 3]
  4:  [2, 4]
  5:  [1, 5]
  6:  [1, 2, 3, 6]
  7:  [1, 7]
  8:  [4, 8]
  9:  [3, 9]
  10: [1, 2, 5, 10]
  11: [1, 11]
  12: [2, 4, 6, 12]
  13: [1, 13]
  14: [1, 2, 7, 14]
  15: [1, 3, 5, 15]
  16: [8, 16]
  17: [1, 17]
  18: [3, 6, 9, 18]
  19: [1, 19]
  20: [2, 4, 10, 20]

Cf. A001615 (row sums), A003557, A005117, A034444 (row lengths), A206778.

for (n=1, 28, fordiv (n, d, if (issquarefree(n/d), print1 (d ", "))))

T(n, k) = n / A206778(n, A034444(n) - k + 1) for any n > 0 and k such that 1 <= k <= A034444(n).

cp's OEIS Frontend

A050328 Number of ordered factorizations of n into squarefree numbers > 1.

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A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

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

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A291109 Numbers that are not the sum of the squarefree divisors of some natural number.

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A295908 Triangle in which n-th row lists divisors d of n such that n/d is squarefree.

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