A293234 -id:A293234 - cp's OEIS frontend

A347708 Number of distinct possible alternating products of odd-length factorizations of n.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 1, 2, 1, 4, 2, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 3, 1, 2, 1, 3, 2

Offset: 1

Gus Wiseman, Oct 11 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.

			Representative factorizations for each of the a(180) = 7 alternating products:
  (2*2*3*3*5) -> 5
     (2*2*45) -> 45
     (2*3*30) -> 20
     (2*5*18) -> 36/5
     (2*9*10) -> 20/9
     (3*4*15) -> 45/4
        (180) -> 180

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A028310, reverse A347707.
Positions of 1's appear to be A037143 \ {1}.
The even-length version for n > 1 is A072670, strict A211159.
Counting only integers appears to give A293234, with evens A046951.
This is the odd-length case of A347460, reverse A038548.
The any-length version for partitions is A347461, reverse A347462.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A276024 counts distinct positive subset-sums of partitions.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
A347050 = factorizations w/ an alternating permutation, complement A347706.
A347441 counts odd-length factorizations with integer alternating product.
Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],OddQ[Length[#]]&]]],{n,100}]

altprod(facs) = prod(i=1,#facs,facs[i]^((-1)^(i-1)));
A347708aux(n, m=n, facs=List([])) = if(1==n, if((#facs)%2, altprod(facs), 0), my(newfacs, r, rats=List([])); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); r = A347708aux(n/d, d, newfacs); if(r, rats = concat(rats,r)))); (rats));
A347708(n) = if(1==n,0,#Set(A347708aux(n))); \\ Antti Karttunen, Jan 29 2025

Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A293227 a(n) is the number of proper divisors of n that are squarefree.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 7, 1, 2, 3, 3, 3, 4, 1, 3, 3, 4, 1, 7, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 8, 1, 3, 4, 2, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 4, 4, 3, 7, 1, 4, 2, 3, 1, 8, 3, 3, 3, 4, 1, 8, 3, 4, 3, 3, 3, 4, 1, 4, 4, 4, 1, 7, 1, 4, 7

Offset: 1

Antti Karttunen, Oct 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A005117, A008683, A008966, A034444, A293228, A293234.

Cf. A001620, A306016.

with(numtheory): seq(coeff(series(add(mobius(k)^2*x^(2*k)/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Oct 28 2018

Table[Count[Most[Divisors[n]],?SquareFreeQ],{n,110}] (* _Harvey P. Dale, Jun 15 2021 *)
a[n_] := 2^PrimeNu[n] - Boole[SquareFreeQ[n]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

PARI
```
A293227(n) = sumdiv(n, d, (d
				
```

a(n) = Sum_{d|n, dA008966(d).
a(n) = A034444(n) - A008966(n).
a(n) = 2^A001221(n) - A008683(n)^2 = 2^omega(n) - mu(n)^2.
G.f.: Sum_{k>=1} mu(k)^2*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Oct 28 2018
Sum_{k=1..n} a(k) ~ (6/Pi^2)*n*(log(n) + 2*(gamma - 1 - zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A293235 a(n) is the sum of proper divisors of n that are square.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 14, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 1, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 21, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 21, 1, 50, 10, 30, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen, Oct 08 2017

nonn

a(n) = 1 if and only if n > 1 is squarefree or the square of a prime. - Robert Israel, Oct 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A010052, A035316, A078434, A293228, A293234.

A035316:= n -> mul((p[1]^(p[2]+2-(p[2] mod 2))-1)/(p[1]^2-1), p = ifactors(n)[2]):
f:= n -> A035316(n) - `if`(issqr(n),n,0):
map(f, [$1..100]); # Robert Israel, Oct 08 2017

Table[Total[Select[Most[Divisors[n]],IntegerQ[Sqrt[#]]&]],{n,120}] (* Harvey P. Dale, Dec 29 2023 *)

PARI
```
A293235(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA010052(d)*d.
a(n) = A035316(n) - (A010052(n)*n).
G.f.: Sum_{k>=1} k^2 * x^(2*k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 13 2021
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3/2)-1)/3 = 0.537458449561... . - Amiram Eldar, Dec 01 2023

A343407 Number of proper divisors of n that are triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 1, 5, 1, 1, 3, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 5, 1, 1, 2, 1, 1, 6, 1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 4

Offset: 1

Ilya Gutkovskiy, Apr 14 2021

nonn

Cf. A000217, A007862, A010054, A032741, A293234, A343408.

a:= n-> add(`if`(issqr(8*d+1), 1, 0), d=numtheory[divisors](n) minus {n}):
seq(a(n), n = 1..105);  # Alois P. Heinz, Apr 14 2021

nmax = 105; CoefficientList[Series[Sum[x^(k (k + 1))/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[If[d < n && IntegerQ[Sqrt[8 d + 1]], 1, 0], {d, Divisors[n]}], {n, 105}]

a(n) = sumdiv(n, d, if ((dMichel Marcus, Apr 14 2021

G.f.: Sum_{k>=1} x^(k*(k+1)) / (1 - x^(k*(k+1)/2)).

a(n) = Sum_{d|n, d < n} A010054(d).

cp's OEIS Frontend

A347708 Number of distinct possible alternating products of odd-length factorizations of n.

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A293227 a(n) is the number of proper divisors of n that are squarefree.

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A293235 a(n) is the sum of proper divisors of n that are square.

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Maple

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A343407 Number of proper divisors of n that are triangular numbers.

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