A072048 -id:A072048 - cp's OEIS frontend

A072047 Number of prime factors of the squarefree numbers: omega(A005117(n)).

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

Offset: 1

Reinhard Zumkeller, Jun 09 2002

For n > 1: length of row n in A265668. - Reinhard Zumkeller, Dec 13 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Cf. A072048.

Cf. A001221, A001222, A013661, A005117, A265668.

a072047 n = a072047_list !! (n-1)
a072047_list = map a001221 $ a005117_list
-- Reinhard Zumkeller, Aug 08 2011

PrimeOmega[Select[Range[200],SquareFreeQ]] (* Harvey P. Dale, May 14 2011 *)

apply(omega, select(issquarefree, [1..200])) \\ Michel Marcus, Nov 25 2022

from math import isqrt
from sympy import mobius, primenu
def A072047(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return primenu(bisection(f)) # Chai Wah Wu, Aug 31 2024

a(n) = A001221(A005117(n)) = A001222(A005117(n)).

Sum_{A005117(k) <= x} a(k) = (1/zeta(2))*x*log(log(x)) + O(x) (Jakimczuk and Lalín, 2022). - Amiram Eldar, Feb 18 2023, corrected Sep 21 2024

A124859 Multiplicative with p^e -> primorial(e), p prime and e > 0.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 30, 6, 4, 2, 12, 2, 4, 4, 210, 2, 12, 2, 12, 4, 4, 2, 60, 6, 4, 30, 12, 2, 8, 2, 2310, 4, 4, 4, 36, 2, 4, 4, 60, 2, 8, 2, 12, 12, 4, 2, 420, 6, 12, 4, 12, 2, 60, 4, 60, 4, 4, 2, 24, 2, 4, 12, 30030, 4, 8, 2, 12, 4, 8, 2, 180, 2, 4, 12, 12, 4, 8, 2, 420, 210, 4, 2, 24, 4, 4

Offset: 1

Reinhard Zumkeller, Nov 10 2006

			From _Michael De Vlieger_, Mar 06 2017: (Start)
a(2) = 2 since 2 = 2^1, thus primorial p_1# = 2.
a(4) = 6 since 4 = 2^2, thus primorial p_2# = 2*3 = 6.
a(6) = 4 because 6 is squarefree with omega(6)=2, thus 2^2 = 4.
a(8) = 30 since 8 = 2^3, thus primorial p_3# = 2*3*5 = 30.
a(10) = 4 since 10 is squarefree with omega(10)=2, thus 2^2 = 4.
a(12) = 12 since 12 = 2^1 * 3^2, thus primorials p_1# * p_2# = 2*6 = 12.
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..5000 (first 1000 terms from R. Zumkeller)
Eric Weisstein's World of Mathematics, Prime Factorization
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A000079, A001221, A001222, A002110, A005117, A028234, A046523, A067029, A072048, A108951, A156552, A181819, A181821, A238745, A278159, A278222.

A124859 := proc(n)
    local a,pf;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*A002110(pf[2]) ;
    end do:
    a ;
end proc:
seq(A124859(n),n=1..80) ; # R. J. Mathar, Oct 06 2017

Table[Which[n == 1, 1, SquareFreeQ@ n, 2^PrimeNu@ n, True, Times @@ Map[Times @@ Prime@ Range@ # &, #[[All, -1]]]] &@ FactorInteger@ n, {n, 86}] (* Michael De Vlieger, Mar 06 2017 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = prod(j=1, f[k,2], prime(j)); f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 16 2015

from sympy.ntheory.factor_ import core
from sympy import factorint, primorial, primefactors
from operator import mul
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    f=factorint(n)
    return n if n<3 else 2**omega(n) if core(n) == n else reduce(mul, [primorial(f[i]) for i in f]) # Indranil Ghosh, May 13 2017

(define (A124859 n) (cond ((= 1 n) 1) (else (* (A002110 (A067029 n)) (A124859 (A028234 n)))))) ;; Antti Karttunen, Mar 06 2017

a(A000040(x)^n) = A002110(n); a(A002110(n)) = A000079(n);
a(A005117(n)) = 2^A001221(A005117(n)) = A072048(n);
A001221(a(n)) = A051903(n); A001222(a(n)) = A001222(n).
From Antti Karttunen, Mar 06 2017: (Start)
a(1) = 1, for n > 1, a(n) = A002110(A067029(n)) * a(A028234(n)).
a(n) = A278159(A156552(n)).
a(A278159(n)) = A278222(n).
a(a(n)) = A046523(n). [after Matthew Vandermast's May 19 2012 formula for the latter sequence]
A181819(a(n)) = A238745(n). [after Matthew Vandermast's formula for the latter sequence]
(End)
a(n) = A108951(A181819(n)). [Primorial inflation of the prime shadow of n] - Antti Karttunen, Sep 15 2023

A366438 The number of divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 8, 4, 4, 2, 8, 2, 6, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 2, 4, 2, 8, 4, 8, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 12, 2, 2, 8, 2, 8, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 16, 4

Offset: 1

Amiram Eldar, Oct 10 2023

1 is the only odd term in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A268335, A366439.

Similar sequences: A048691, A072048, A076400, A358040, A363194, A363195.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], Times @@ (e + 1), Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(vecprod(apply(x -> x+1, e)), ", ")));

from math import prod
from itertools import count, islice
from sympy import factorint
def A366438_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e&1 for e in f):
            yield prod(e+1 for e in f)
A366438_list = list(islice(A366438_gen(),30)) # Chai Wah Wu, Oct 10 2023

a(n) = A000005(A268335(n)).

A358040 a(n) is the number of divisors of the n-th cubefree number.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 4, 2, 6, 2, 4, 4, 2, 6, 2, 6, 4, 4, 2, 3, 4, 6, 2, 8, 2, 4, 4, 4, 9, 2, 4, 4, 2, 8, 2, 6, 6, 4, 2, 3, 6, 4, 6, 2, 4, 4, 4, 2, 12, 2, 4, 6, 4, 8, 2, 6, 4, 8, 2, 2, 4, 6, 6, 4, 8, 2, 4, 2, 12, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 2, 6, 6, 9, 2

Offset: 1

Amiram Eldar, Oct 29 2022

nonn

The analogous sequence with squarefree numbers is A072048.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zhu Weiyi, On the cube free number sequences, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.

Cf. A000005, A001620 (gamma), A004709, A072048, A073002 (-zeta'(2)), A147533 (2*gamma-1), A358039.

DivisorSigma[0, Select[Range[100], Max[FactorInteger[#][[;;, 2]]] < 3 &]]

from sympy import mobius, integer_nthroot, divisor_count
def A358040(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_count(m) # Chai Wah Wu, Aug 06 2024

a(n) = A000005(A004709(n)).

Sum_{k=1..n} a(k) = (36*c_1/Pi^4) * n * (log(n) + (2*gamma - 1) - 24*zeta'(2)/Pi^2 - 4*c_2) + O(n^(1/2 + eps)), where c_1 = Product_{p prime} ((p^2+2*p+3)/(p+1)^2) = 1.58095136661854869148023... and c_2 = Sum_{p prime} p*log(p)/((p+1)*(p^2+2*p+3)) = 0.229224... (Weiyi, 2004).

A363194 Number of divisors of the n-th powerful number A001694(n).

Original entry on oeis.org

1, 3, 4, 3, 5, 3, 4, 6, 9, 3, 7, 12, 5, 9, 12, 3, 4, 8, 15, 3, 9, 12, 16, 9, 6, 9, 18, 3, 15, 4, 3, 12, 15, 20, 9, 9, 12, 10, 3, 21, 5, 20, 12, 9, 7, 15, 18, 3, 24, 27, 3, 12, 18, 16, 11, 9, 12, 24, 9, 9, 25, 12, 4, 12, 3, 12, 9, 9, 18, 21, 3, 28, 27, 36, 3, 15

Offset: 1

Amiram Eldar, May 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).

Cf. A000005, A001694, A180114, A306458, A343443.

Similar sequences: A072048, A076400, A363195.

DivisorSigma[0, Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

apply(numdiv, select(ispowerful, [1..3000]))

from itertools import count, islice
from math import prod
from sympy import factorint
def A363194_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>1 for e in f):
            yield prod(e+1 for e in f)
A363194_list = list(islice(A363194_gen(),20)) # Chai Wah Wu, May 21 2023

a(n) = A000005(A001694(n)).
Sum_{A001694(k) < x} a(k) = c_1 * sqrt(x) * log(x)^2 + c_2 * sqrt(x) * log(x) + c_3 * sqrt(x) + O(x^(5/12 + eps)), where c_1, c_2 and c_3 are constants. c_1 = Product_{p prime} (1 + 4/p^(3/2) - 1/p^2 - 6/p^(5/2) + 2/p^(7/2))/8 = 0.516273682988566836609... . [corrected Sep 21 2024]
a(n) = A343443(A306458(n)). - Amiram Eldar, Sep 01 2023

A366536 The number of unitary divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 4, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 8, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 8, 4, 4, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8

Offset: 1

Amiram Eldar, Oct 12 2023

The number of unitary divisors of the squarefree numbers (A005117) is the same as the number of divisors of the squarefree numbers (A072048), because all the divisors of a squarefree number are unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004709, A005117, A034444, A072048, A077610, A358040, A366440, A366537.

Similar sequences: A366534, A366538.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # < 3 &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], iscubefree = 1); for(i = 1, #e, if(e[i] > 2, iscubefree = 0; break)); if(iscubefree, print1(2^(#e), ", ")));

from sympy.ntheory.factor_ import udivisor_count
from sympy import mobius, integer_nthroot
def A366536(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return udivisor_count(m) # Chai Wah Wu, Aug 05 2024

a(n) = A034444(A004709(n)).

A363195 Number of divisors of the n-th cubefull number A036966(n).

Original entry on oeis.org

1, 4, 5, 4, 6, 7, 5, 4, 8, 16, 6, 9, 4, 20, 10, 5, 20, 7, 24, 16, 11, 25, 4, 28, 24, 20, 12, 8, 4, 5, 30, 16, 6, 16, 32, 30, 24, 13, 4, 20, 35, 20, 28, 9, 4, 36, 36, 28, 14, 16, 25, 20, 40, 16, 24, 35, 4, 40, 5, 42, 7, 32, 15, 6, 20, 32, 16, 20, 10, 30, 45, 20

Offset: 1

Amiram Eldar, May 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).

Cf. A000005, A036966, A362986.

Similar sequences: A072048, A076400, A363194.

DivisorSigma[0, Select[Range[25000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(numdiv(k), ", ")));

from itertools import count, islice
from math import prod
from sympy import factorint
def A363195_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>2 for e in f):
            yield prod(e+1 for e in f)
A363195_list = list(islice(A363195_gen(),20)) # Chai Wah Wu, May 21 2023

a(n) = A000005(A036966(n)).

Sum_{A036966(k) < x} a(k) = c_1 * x^(1/3) * log(x)^3 + c_2 * x^(1/3) * log(x)^2 + c_3 * x^(1/3) * log(x) + c_4 * x^(1/3) + O(x^(7/24 + eps)), where c_1, c_2, c_3 and c_4 are constants. c_1 = Product_{p prime} ((1-1/p)^4 * (1 + 1/((p^(1/3) - 1)^2*p^(1/3)) + 3/(p-p^(2/3))))/162 = 0.1346652397135839416... . [corrected Sep 21 2024]

A073245 Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number.

Original entry on oeis.org

1, 6, 12, 30, 72, 56, 180, 132, 182, 336, 360, 306, 380, 672, 792, 552, 1092, 870, 2160, 992, 1584, 1836, 1680, 1406, 2280, 2184, 1722, 4032, 1892, 3312, 2256, 3672, 2862, 3960, 4560, 5220, 3540, 3782, 5952, 5460, 9504, 4556, 6624, 10080, 5112, 5402

Offset: 1

Reinhard Zumkeller, Jul 21 2002

nonn

			14 is the 10th squarefree number: A005117(10)=14=2*7, the cubefree numbers with squarefree kernel =14 are 14, 28=2*2*7, 98=2*7*7 and 196=2*2*7*7; therefore a(10)=14+28+98+196=336; a(10)=A062822(10)*A005117(10)=24*14=336.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004709, A005117, A007947, A062822, A064987, A072048, A306633.

Cf. A002117, A013661.

Map[# * DivisorSigma[1, #] &, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 14 2020 *)

apply(x->(x*sigma(x)), select(issquarefree, [1..100])) \\ Michel Marcus, Oct 18 2020

from math import isqrt
from sympy import mobius, divisor_sigma
def A073245(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m*divisor_sigma(m) # Chai Wah Wu, Aug 12 2024

a(n) = A062822(n)*A005117(n).
Sum_{n>=1} 1/a(n) = A306633. - Amiram Eldar, Oct 14 2020
a(n) = A064987(A005117(n)). - Michel Marcus, Oct 18 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)^3/(3*zeta(3)) = 1.23423882415851340020... . - Amiram Eldar, Oct 09 2023

A366441 The number of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 3, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 2, 3, 2, 6, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4

Offset: 1

Amiram Eldar, Oct 10 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001620, A007310, A366442.

Similar sequences: A048691, A072048, A076400, A099774, A358040, A363194, A363195.

a[n_] := DivisorSigma[0, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI
```
a(n) = numdiv((3*n)\2 << 1 - 1)
```

from sympy import divisor_count
def A366441(n): return divisor_count((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

a(n) = A000005(A007310(n)).

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1 + 2*log(6)) * n / 3, where gamma is Euler's constant (A001620).

cp's OEIS Frontend

A072047 Number of prime factors of the squarefree numbers: omega(A005117(n)).

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A124859 Multiplicative with p^e -> primorial(e), p prime and e > 0.

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A366438 The number of divisors of the exponentially odd numbers (A268335).

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A358040 a(n) is the number of divisors of the n-th cubefree number.

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A363194 Number of divisors of the n-th powerful number A001694(n).

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A366536 The number of unitary divisors of the cubefree numbers (A004709).

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A363195 Number of divisors of the n-th cubefull number A036966(n).

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