A183096 -id:A183096

A304326

Number of ways to write n as a product of a number that is not a perfect power and a squarefree number.

+10
11

0, 1, 1, 1, 1, 3, 1, 0, 1, 3, 1, 3, 1, 3, 3, 0, 1, 3, 1, 3, 3, 3, 1, 2, 1, 3, 0, 3, 1, 7, 1, 0, 3, 3, 3, 3, 1, 3, 3, 2, 1, 7, 1, 3, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 7, 1, 3, 3, 0, 3, 7, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 3, 7, 1, 2, 0, 3, 1, 7, 3, 3, 3, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

EXAMPLE

The a(180) = 7 ways are (6*30), (12*15), (18*10), (30*6), (60*3), (90*2), (180*1).

MATHEMATICA

radQ[n_]:=And[n>1, GCD@@FactorInteger[n][[All, 2]]===1];

Table[Length[Select[Divisors[n], radQ[#]&&SquareFreeQ[n/#]&]], {n, 100}]

PROG

(PARI) a(n)={sumdiv(n, d, d<>1 && !ispower(d) && issquarefree(n/d))} \\ Andrew Howroyd, Aug 26 2018

CROSSREFS

Positions of zeros are A246549. Range appears to be A075427.

Cf. A000961, A001055, A001597, A001694, A005117, A007916, A034444, A091050, A183096, A303386, A303707, A304327, A304328.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 10 2018

STATUS

approved

A304327

Number of ways to write n as a product of a perfect power and a squarefree number.

+10
10

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

First term greater than 2 is a(746496) = 3.

LINKS

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

EXAMPLE

The a(746496) = 3 ways are 12^5*3, 72^3*2, 864^2*1.

MATHEMATICA

Table[Length[Select[Divisors[n], (#===1||GCD@@FactorInteger[#][[All, 2]]>1)&&SquareFreeQ[n/#]&]], {n, 100}]

PROG

(PARI) A304327(n) = sumdiv(n, d, issquarefree(n/d)*((1==d)||ispower(d))); \\ Antti Karttunen, Jul 29 2018

CROSSREFS

Cf. A000961, A001055, A001597, A005117, A007916, A034444, A091050, A183096, A203025, A246549, A303386, A304326, A304328.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 10 2018

EXTENSIONS

More terms from Antti Karttunen, Jul 29 2018

STATUS

approved

A304817

Number of divisors of n that are either 1 or not a perfect power.

+10
7

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 11, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 11, 4, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

First differs from A183095 at a(80) = 8, A183095(80) = 7.

LINKS

Table of n, a(n) for n=1..86.

FORMULA

a(n) = A183096(n) + 1.

EXAMPLE

The a(72) = 8 divisors of 72 that are either 1 or not a perfect power are {1, 2, 3, 6, 12, 18, 24, 72}. Missing are {4, 8, 9, 36}.

MATHEMATICA

Table[DivisorSum[n, Boole[GCD@@FactorInteger[#][[All, 2]]==1]&], {n, 100}]

PROG

(PARI) a(n) = sumdiv(n, d, !ispower(d)); \\ Michel Marcus, May 19 2018

CROSSREFS

Cf. A000005, A001597, A007916, A091050, A183096, A304326, A304362, A304653, A304779, A304819, A304820.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 18 2018

STATUS

approved

A293524

a(n) = Product_{d|n, d>1} prime(A052409(d)).

+10
6

1, 2, 2, 6, 2, 8, 2, 30, 6, 8, 2, 48, 2, 8, 8, 210, 2, 48, 2, 48, 8, 8, 2, 480, 6, 8, 30, 48, 2, 128, 2, 2310, 8, 8, 8, 864, 2, 8, 8, 480, 2, 128, 2, 48, 48, 8, 2, 6720, 6, 48, 8, 48, 2, 480, 8, 480, 8, 8, 2, 3072, 2, 8, 48, 30030, 8, 128, 2, 48, 8, 128, 2, 17280, 2, 8, 48, 48, 8, 128, 2, 6720, 210, 8, 2, 3072, 8, 8, 8, 480

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(n) = Product_{d|n, d>1} A000040(A052409(d)).

Other identities. For all n >= 1:

A001222(a(n)) = A032741(n).

A007814(a(n)) = A183096(n).

A064989(a(n)) = A294875(n).

PROG

(PARI) A293524(n) = { my(m=1, e); fordiv(n, d, if(d>1, e = ispower(d); if(!e, m += m, m *= prime(e)))); m; };

CROSSREFS

Cf. A000040, A052409, A294873, A294874, A294875.

Cf. also A293514, A293442.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 11 2017

STATUS

approved

A294873

a(n) = Product_{d|n, d>1, d = x^(2k-1) for some maximal k >= 1} prime(k).

+10
5

1, 2, 2, 2, 2, 8, 2, 6, 2, 8, 2, 16, 2, 8, 8, 6, 2, 16, 2, 16, 8, 8, 2, 96, 2, 8, 6, 16, 2, 128, 2, 30, 8, 8, 8, 32, 2, 8, 8, 96, 2, 128, 2, 16, 16, 8, 2, 192, 2, 16, 8, 16, 2, 96, 8, 96, 8, 8, 2, 1024, 2, 8, 16, 30, 8, 128, 2, 16, 8, 128, 2, 384, 2, 8, 16, 16, 8, 128, 2, 192, 6, 8, 2, 1024, 8, 8, 8, 96, 2, 1024, 8, 16, 8, 8, 8, 1920, 2, 16, 16, 32, 2, 128, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(n) = Product_{d|n, d>1, r = A052409(d) is odd} A000040((r+1)/2).

Other identities. For all n >= 1:

A001222(a(n)) = A056595(n).

A007814(a(n)) = A183096(n).

PROG

(PARI) A294873(n) = { my(m=1, e); fordiv(n, d, if(d>1, e = ispower(d); if(!e, m += m, if((e>1)&&(e%2), m *= prime((e+1)/2))))); m; };

CROSSREFS

Cf. A000040, A056595, A052409, A183096.

Cf. A293524, A294874, A294875.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 11 2017

STATUS

approved

A183093

a(1) = 0; thereafter, a(n) = number of divisors d of n such that if d = Product_(i) (p_i^e_i) then all e_i <= 1.

+10
4

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 6, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 10, 3, 3, 3, 5, 1, 10, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

a(n) = number of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >=2.

Sequence is not the same as A183093(n): a(72) = 6, A183093(72) = 7.

LINKS

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

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(n) = A000005(n) - A005361(n) = A183095(n) - 1.

a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k - 1, a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

EXAMPLE

For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.

PROG

(Scheme) (define (A183093 n) (- (A000005 n) (A005361 n))) ;; Antti Karttunen, Nov 23 2017

CROSSREFS

Cf. A000005, A005361, A183094, A183095, A183096.

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Dec 25 2010

STATUS

approved

A304649

Number of divisors d|n such that neither d nor n/d is a perfect power greater than 1.

+10
2

1, 2, 2, 1, 2, 4, 2, 0, 1, 4, 2, 4, 2, 4, 4, 0, 2, 4, 2, 4, 4, 4, 2, 4, 1, 4, 0, 4, 2, 8, 2, 0, 4, 4, 4, 5, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 1, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 10, 2, 4, 4, 0, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 4, 0, 4, 2, 10, 4, 4, 4, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

Index entries for sequences computed from exponents in factorization of n

EXAMPLE

The a(36) = 5 ways to write 36 as a product of two numbers that are not perfect powers greater than 1 are 2*18, 3*12, 6*6, 12*3, 18*2.

MATHEMATICA

nn=1000;

sradQ[n_]:=GCD@@FactorInteger[n][[All, 2]]===1;

Table[Length@Select[Divisors[n], sradQ[n/#]&&sradQ[#]&], {n, nn}]

PROG

(PARI) a(n) = sumdiv(n, d, !ispower(d) && !ispower(n/d)); \\ Michel Marcus, May 17 2018

CROSSREFS

Cf. A000005, A001055, A007427, A007916, A034444, A045778, A162247, A183096, A281116, A301700, A303386, A303707, A304650.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 15 2018

STATUS

approved

A304650

Number of ways to write n as a product of two positive integers, neither of which is a perfect power.

+10
1

0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 0, 2, 0, 6, 0, 0, 2, 2, 2, 5, 0, 2, 2, 2, 0, 6, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0, 8, 0, 2, 2, 0, 2, 6, 0, 2, 2, 6, 0, 4, 0, 2, 2, 2, 2, 6, 0, 2, 0, 2, 0, 8, 2, 2, 2, 2, 0, 8, 2, 2, 2, 2, 2, 2, 0, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Table of n, a(n) for n=1..98.

EXAMPLE

The a(60) = 8 ways to write 60 as a product of two numbers, neither of which is a perfect power, are 2*30, 3*20, 5*12, 6*10, 10*6, 12*5, 20*3, 30*2.

MATHEMATICA

radQ[n_]:=And[n>1, GCD@@FactorInteger[n][[All, 2]]===1];

Table[Length[Select[Divisors[n], radQ[#]&&radQ[n/#]&]], {n, 100}]

PROG

(PARI) ispow(n) = (n==1) || ispower(n);

a(n) = sumdiv(n, d, !ispow(d) && !ispow(n/d)); \\ Michel Marcus, May 17 2018

CROSSREFS

Cf. A000005, A001055, A007427, A007916, A034444, A045778, A162247, A183096, A281116, A301700, A303386, A303707, A304649.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 15 2018

STATUS

approved