A190108 -id:A190108 - 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

A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Matthew Vandermast, May 08 2014

nonn

Values of n for which all numbers in row A238747(n) are even. Also, numbers n such that A000005(n^m) is a perfect square for all nonnegative integers m; numbers n such that A181819(n) is a perfect square; numbers n such that A182860(n) is odd.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 33, 314, 3119, 31436, 315888, 3162042, 31626518, 316284320, 3162915907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3162... . - Amiram Eldar, Nov 28 2023

			The prime signature of 36 = 2^2 * 3^2 is {2,2}. One distinct integer (namely, 2) appears in the prime signature, and it appears an even number of times (2 times). Hence, 36 appears in the sequence.
The prime factorization of 1260 = 2^2 * 3^2 * 5^1 * 7^1. Exponent 2 occurs twice (an even number of times), as well as exponent 1, thus 1260 is included. It is also the first term k > 1 in this sequence for which A182850(k) = 4, not 3. - _Antti Karttunen_, Feb 06 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequences include A006881, A030229, A046386, A077448, A162142, A179690, A190108, A190465, A367590.
Subsequence of A000379, A028260, A063774 and A268390.
Cf. A000005, A181819, A182850, A182860, A238747.

q[n_] := n == 1 || AllTrue[Tally[FactorInteger[n][[;; , 2]]][[;; , 2]], EvenQ]; Select[Range[200], q] (* Amiram Eldar, Nov 28 2023 *)

is(n) = {my(e = factor(n)[, 2], m = #e); if(m%2, return(0)); e = vecsort(e); forstep(i = 1, m, 2, if(e[i] != e[i+1], return(0))); 1;} \\ Amiram Eldar, Nov 28 2023

(define A238748 (MATCHING-POS 1 1 (lambda (n) (square? (A181819 n)))))
(define (square? n) (not (zero? (A010052 n))))
;; Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 06 2016

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

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

A190377 Numbers with prime factorization p^2q^2r^2*s^2 where p, q, r, and s are distinct primes.

Original entry on oeis.org

44100, 108900, 152100, 213444, 260100, 298116, 324900, 476100, 509796, 592900, 636804, 736164, 756900, 828100, 864900, 933156, 1232100, 1258884, 1334025, 1416100, 1483524, 1512900, 1572516, 1664100, 1695204, 1758276, 1768900, 1863225

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 09 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of prime signatures, 2010.
Index to sequences related to prime signature.

Cf. A190108, A190109.

Cf. A046386, A085548, A085964, A085966, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,2,2};Select[Range[3000000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\900), t1=p^2; forprime(q=2,sqrtint(lim\(36*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(4*t2)), if(r==p || r==q, next); t3=r^2*t2; forprime(s=2,sqrtint(lim\t3), if(s==p || s==q || s==r, next); listput(v, t3*s^2))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A190377(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1),1) for b,m in enumerate(primerange(k+1,integer_nthroot(x//k,3)[0]+1),a+1) for c,r in enumerate(primerange(m+1,isqrt(x//(k*m))+1),b+1)))
    return bisection(f,n,n)**2 # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = (P(2)^4 - 6*P(2)^2*P(4) + 8*P(2)*P(6) + 3*P(4)^2 - 6*P(8))/24 = 0.00010511750849230980748..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

a(n) = A046386(n)^2. - Chai Wah Wu, Mar 27 2025

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

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A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

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

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A190377 Numbers with prime factorization p^2*q^2*r^2*s^2 where p, q, r, and s are distinct primes.

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

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

A190377 Numbers with prime factorization p^2q^2r^2*s^2 where p, q, r, and s are distinct primes.