A190107 -id:A190107 - cp's OEIS frontend

A050326 Number of factorizations of n into distinct squarefree numbers > 1.

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

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).
a(A212164(n)) = 0; a(A212166(n)) = 1; a(A006881(n)) = 2; a(A190107(n)) = 3; a(A085987(n)) = 4; a(A225228(n)) = 5; a(A179670(n)) = 7; a(A162143(n)) = 8; a(A190108(n)) = 11; a(A212167(n)) > 0; a(A212168(n)) > 1. - Reinhard Zumkeller, May 03 2013
The comment that a(A212164(n)) = 0 is incorrect. For example, 3600 belongs to A212164 but a(3600) = 1. The positions of zeros in this sequence are A293243. - Gus Wiseman, Oct 10 2017

			The a(30) = 5 factorizations are: 2*3*5, 2*15, 3*10, 5*6, 30. The a(180) = 5 factorizations are: 2*3*5*6, 2*3*30, 2*6*15, 3*6*10, 6*30. - _Gus Wiseman_, Oct 10 2017

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

Cf. A001055, A005117, A045778, A046523, A050320, A050327, a(p^k)=0 (p>1), a(A002110)=A000110, a(n!)=A103775(n), A206778, A293243.

import Data.List (subsequences, genericIndex)
a050326 n = genericIndex a050326_list (n-1)
a050326_list = 1 : f 2 where
   f x = (if x /= s then a050326 s
                    else length $ filter (== x) $ map product $
                         subsequences $ tail $ a206778_row x) : f (x + 1)
         where s = a046523 x
-- Reinhard Zumkeller, May 03 2013

N:= 1000: # to get a(1)..a(N)
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
     S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
convert(A,list); # Robert Israel, Oct 10 2017

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Oct 10 2017 *)

Dirichlet g.f.: prod{n is squarefree and > 1}(1+1/n^s).

a(n) = A050327(A101296(n)). - R. J. Mathar, May 26 2017

A190110 Numbers with prime factorization pqrst^4 (where p, q, r, s, t are distinct primes).

Original entry on oeis.org

18480, 21840, 28560, 31920, 34320, 38640, 44880, 48048, 48720, 50160, 52080, 53040, 59280, 60720, 62160, 62370, 62832, 68880, 70224, 71760, 72240, 73710, 74256, 76560, 77520, 78960, 80080, 81840, 82992, 85008, 89040, 90480, 93840, 96390, 96720, 97680, 99120

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

That is, numbers with prime signature {1,1,1,1,4}.

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^4)*3*5*7*11 = 18480;
a(2) = (2^4)*3*5*7*13 = 21840;
a(3) = (2^4)*3*5*7*17 = 28560;
a(4) = (2^4)*3*5*7*19 = 31920.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime signature.
Wikipedia, Prime signature.
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A179672, A190107, A190108, A190109.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,1,4};Select[Range[150000],f]

list(lim)=my(v=List(),t1,t2,t3,t4); forprime(p1=2,sqrtnint(lim\210, 4), t1=p1^4; forprime(p2=2,lim\(30*t1), if(p2==p1, next); t2=p2*t1; forprime(p3=2,lim\(6*t2), if(p3==p1 || p3==p2, next); t3=p3*t2; forprime(p4=2,lim\(2*t3), if(p4==p1 || p4==p2 || p4==p3, next); t4=p4*t3; forprime(p5=2,lim\t4, if(p5==p1 || p5==p2 || p5==p3 || p5==p4, next); listput(v, t4*p5)))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Name edited by Petros Hadjicostas, Oct 26 2019

A190108 Numbers with prime factorization pqr^3*s^3 (where p, q, r, s are distinct primes).

Original entry on oeis.org

7560, 11880, 14040, 16632, 18360, 19656, 20520, 21000, 24840, 25704, 28728, 30888, 31320, 33000, 33480, 34776, 39000, 39960, 40392, 41160, 43848, 44280, 45144, 46440, 46872, 47250, 47736, 50760, 51000, 53352, 54648, 55944, 57000, 57240, 61992, 63720, 64584

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

A050326(a(n)) = 11. - Reinhard Zumkeller, May 03 2013

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*(3^3)*5*7 = 7560;
a(2) = (2^3)*(3^3)*5*11 = 11880;
a(3) = (2^3)*(3^3)*5*13 = 14040;
a(4) = (2^3)*(3^3)*7*11 = 16632.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A179704, A190106, A190107.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3,3};Select[Range[150000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\120, 3), t1=p^3; forprime(q=2,sqrtnint(lim\(6*t1), 3), if(q==p, next); t2=q^3*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Name edited by Petros Hadjicostas, Oct 26 2019

A190109 Numbers with prime factorization pq^2r^2*s^3 (where p, q, r, s are distinct primes).

Original entry on oeis.org

12600, 17640, 18900, 19800, 23400, 26460, 29400, 29700, 30600, 31500, 34200, 35100, 38808, 41400, 43560, 45864, 45900, 49500, 51300, 52200, 55800, 58212, 58500, 59976, 60840, 60984, 61740, 62100, 65340, 66150, 66600, 67032, 68796, 72600, 73500, 73800, 76500

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

That is, numbers with prime signature {1,2,2,3}.

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*(3^2)*(5^2)*7 = 12600;
a(2) = (2^3)*(3^2)*5*(7^2) = 17640;
a(3) = (2^2)*(3^3)*(5^2)*7 = 18900;
a(4) = (2^3)*(3^2)*(5^2)*11 = 19800.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime signature.
Wikipedia, Prime signature.
Will Nicholes, Prime Signatures.
Index to sequences related to prime signature

Cf. A190106, A190107, A190108.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,2,3};Select[Range[150000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\180, 3), t1=p^3; forprime(q=2,sqrtint(lim\(12*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(2*t2)), if(r==p || r==q, next); t3=r^2*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

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A050326 Number of factorizations of n into distinct squarefree numbers > 1.

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A190110 Numbers with prime factorization p*q*r*s*t^4 (where p, q, r, s, t are distinct primes).

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A190108 Numbers with prime factorization p*q*r^3*s^3 (where p, q, r, s are distinct primes).

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A190109 Numbers with prime factorization p*q^2*r^2*s^3 (where p, q, r, s are distinct primes).

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A190110 Numbers with prime factorization pqrst^4 (where p, q, r, s, t are distinct primes).

A190108 Numbers with prime factorization pqr^3*s^3 (where p, q, r, s are distinct primes).

A190109 Numbers with prime factorization pq^2r^2*s^3 (where p, q, r, s are distinct primes).