A076399 -id:A076399 - cp's OEIS frontend

A076400 Number of divisors of n-th perfect power.

1, 3, 4, 3, 5, 3, 4, 6, 9, 3, 7, 5, 9, 3, 4, 8, 15, 3, 9, 16, 9, 6, 9, 3, 15, 4, 3, 15, 9, 9, 10, 3, 21, 5, 9, 7, 15, 3, 27, 3, 16, 11, 9, 9, 9, 25, 4, 3, 9, 9, 21, 3, 28, 27, 3, 15, 15, 12, 9, 8, 4, 3, 27, 5, 15, 9, 15, 16, 3, 21, 9, 6, 21, 9, 9, 16, 3, 45, 3, 9, 15, 13, 9, 27, 3, 15, 9, 27, 4

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A001597, A076398, A076399, A076401.

DivisorSigma[0, {1}~Join~Select[Range[5000], GCD @@ FactorInteger[#][[All, -1]] > 1 &]] (* Michael De Vlieger, Dec 16 2021 *)

from sympy import mobius, integer_nthroot, divisor_count
def A076400(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return int(divisor_count(kmax)) # Chai Wah Wu, Aug 14 2024

a(n) = A000005(A001597(n)).

A076398 Number of distinct prime factors of n-th perfect power.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A001221, A001597, A025478, A076399, A076400.

a076398 = a001221 . a025478  -- Reinhard Zumkeller, Mar 28 2014

ppQ[1] = True; ppQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; PrimeNu /@ Select[Range[10^4], ppQ] (* Jean-François Alcover, Jul 15 2017 *)

lista(nn) = for(n=1, nn, if ((n==1) || ispower(n), print1(omega(n), ", "))); \\ Michel Marcus, Jul 15 2017

from sympy import mobius, integer_nthroot, primenu
def A076398(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return int(primenu(kmax)) # Chai Wah Wu, Aug 14 2024

a(n) = A001221(A001597(n)).

a(n) = A001221(A025478(n)).

A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 5, 4, 4, 5, 2, 3, 7, 6, 2, 4, 5, 6, 4, 5, 8, 7, 2, 6, 3, 2, 5, 6, 7, 4, 4, 5, 9, 2, 8, 4, 7, 5, 4, 6, 6, 7, 2, 8, 6, 2, 5, 7, 6, 10, 4, 5, 9, 4, 4, 8, 5, 3, 5, 2, 5, 4, 4, 7, 8, 2, 9, 6, 7, 2, 6, 8, 7, 6, 11, 4, 7, 3, 2, 10, 5

Offset: 1

Amiram Eldar, Feb 18 2023

nonn

Amiram Eldar, 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. A001222, A001694, A083342, A090699.

Similar sequences: A072047, A076399.

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

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

a(n) = A001222(A001694(n)).

Sum_{A001694(k) < x} a(k) = (2*zeta(3/2)/zeta(3))*sqrt(x)*log(log(x)) + (2*(B_2 - log(2)) + Sum_{p prime} (3/((p^(3/2)+1))))*(zeta(3/2)/zeta(3))*sqrt(x) + O(sqrt(x)/sqrt(log(x))), where B_2 = A083342 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).

Original entry on oeis.org

0, 3, 4, 3, 5, 6, 4, 3, 7, 6, 5, 8, 3, 7, 9, 4, 7, 6, 8, 6, 10, 8, 3, 9, 8, 7, 11, 7, 3, 4, 9, 6, 5, 6, 10, 9, 8, 12, 3, 7, 10, 7, 9, 8, 3, 11, 10, 9, 13, 6, 8, 7, 11, 6, 8, 10, 3, 12, 4, 11, 6, 10, 14, 5, 7, 10, 6, 7, 9, 9, 12, 7, 9, 11, 3, 8, 9, 13, 7, 4, 3

Offset: 1

Amiram Eldar, May 13 2023

nonn

Amiram Eldar, 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. A001222, A036966, A362974.

Similar sequences: A072047, A076399, A360729.

PrimeOmega[Select[Range[10000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

iscubefull(n) = n==1 || vecmin(factor(n)[, 2]) > 2;
apply(bigomega, select(iscubefull, [1..10000]))

a(n) = A001222(A036966(n)).
a(n) >= 3, for n > 1.
Sum_{A036966(k) < x} a(k) = 3*c*x^(1/3)*log(log(x)) + (3*(B_2 - log(2)) + Sum_{p prime} ((4*p^(1/3)+5)/(p^(5/3)+p^(1/3)+1)))*c*x^(1/3) + O(x^(1/3)/sqrt(log(x))), where B_2 = A083342 and c = A362974 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

cp's OEIS Frontend

A076400 Number of divisors of n-th perfect power.

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A076398 Number of distinct prime factors of n-th perfect power.

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A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

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Author

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Crossrefs

Programs

Mathematica

PARI

Formula

A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula