A382061 -id:A382061 - cp's OEIS frontend

A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

1, 6, 12, 12, 24, 30, 36, 48, 72, 56, 96, 144, 108, 180, 216, 132, 150, 192, 288, 182, 336, 360, 432, 360, 324, 384, 576, 306, 648, 392, 380, 672, 720, 864, 672, 792, 900, 768, 552, 1152, 750, 1296, 1080, 1092, 972, 1344, 1440, 870, 1728, 2160, 992, 1584

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Pi^2/6 (A013661).

The asymptotic density of a sequence S that possesses the property that an integer k is a term if and only if its powerful part, A057521(k) is a term, is (1/zeta(2)) * Sum_{n>=1, A001694(n) is a term of S} 1/a(n). Examples for such sequences are the e-perfect numbers (A054979), the exponential abundant numbers (A129575), and other sequences listed in the Crossrefs section. - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A001615, A001694, A013661, A082695, A112526.

Sequences whose density can be calculated using this sequence: A054979, A129575, A307958, A308053, A321147, A322858, A323310, A328135, A339936, A340109, A364990, A382061, A383693, A383695, A383697.

psi[1]=1; psi[n_] := n * Times@@(1+1/Transpose[FactorInteger[n]][[1]]); psi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after T. D. Noe at A001615 and Harvey P. Dale at A001694 *)

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A323332(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+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
    def f(x):
        c, l = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    a = primefactors(m:=bisection(f,n,n))
    return m*prod(p+1 for p in a)//prod(a) # Chai Wah Wu, Sep 14 2024

A382063 Numbers whose number of coreful divisors is divisible by their number of exponential divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Amiram Eldar, Mar 14 2025

First differs from A377019 at n = 55: A377019(55) = 64 is not a term of this sequence.
First differs from A344742 at n = 62: A344742(62) = 72 is not a term of this sequence.
All the cubefree numbers (A004709) are terms. The least term that is not cubefree is a(215) = 256 = 2^8. The cubefree numbers are the terms whose number of coreful divisors is equal to their number of exponential divisors.
All the exponentially refactorable numbers (A382065) are terms. The least term that is not in A382065 is a(362) = 432 = 2^4 * 3^3. The next terms that are not in A382065 are 648, 2000, 2160, 3024, 3240, 4536, 4752, 5000, ... .
For a number k whose prime factorization is Product_{i} p_i^e_i, a coreful divisor d of k has the prime factorization Product_{i} p_i^f_i with f_i >= 1 for all i. An exponential divisor of k is a coreful divisor with the additional condition that f_i | e_i for all i.
Numbers k such that A049419(k) | A005361(k).
The criterion according to which a number belongs to this sequence depends only on the prime signature of this number: if {e_1, e_2, ... } are the exponents in the prime factorization of k then k is a term if and only if A005361(k)/A049419(k) = Product_{i} e_i/A000005(e_i) is an integer.
A number k is a term if and only if the cubefull part of k, A360540(k), is a term. Therefore, the primitive terms of this sequence are the cubefull terms, A382064.
The asymptotic density of this sequence is Sum_{n>=1} f(A382064(n)) = 0.83697905945047..., where f(n) = (1/(zeta(3)*n)) * Product_{prime p|n} (p^2/(p^2+p+1)).

			2 is a term since A005361(2) = A049419(2) = 2, so 2 | 2.
256 is a term since A005361(256) = 8, A049419(256) = 4, and 4 | 8.

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

Cf. A000005, A002117, A005361, A036966, A049419, A344742, A360540, A377019, A382061.

Subsequences: A004709, A382064, A382065.

q[k_] := Module[{e = FactorInteger[k][[;;, 2]]}, Divisible[Times @@ e, Times @@ DivisorSigma[0, e]]]; Select[Range[100], # == 1 || q[#] &]

isok(k) = {my(e = factor(k)[, 2]); !(vecprod(e) % vecprod(apply(x -> numdiv(x), e)));}

A382062 Powerful numbers whose number of divisors is divisible by their number of unitary divisors.

Original entry on oeis.org

1, 8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 343, 392, 432, 500, 512, 648, 675, 864, 968, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1728, 1944, 2000, 2048, 2187, 2197, 2312, 2744, 2888, 3087, 3125, 3200, 3267, 3375, 3456, 4000, 4232, 4563, 4913, 5000, 5324, 5400

Offset: 1

Amiram Eldar, Mar 14 2025

Powerful numbers k such that A034444(k) | A000005(k).

The primitive terms of A382061: if k is a term and m is a squarefree number that is coprime to k, then k*m is a term of A382061. The asymptotic density of A382061 can be calculated using the terms of this sequence (see A382061 for a formula).

			27 = 3^3 is a term since it is powerful, A000005(27) = 4, A034444(27) = 2, and 2 | 4.
72 = 2^3 * 3^2 is a term since it is powerful, A000005(72) = 12, A034444(72) = 4, and 4 | 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A382061.

Cf. A000005, A034444, A382064.

q[k_] := Module[{e = FactorInteger[k][[;;, 2]]}, AllTrue[e, # > 1 &] && Divisible[Times @@ (e+1), 2^Length[e]]]; Select[Range[5400], # == 1 || q[#] &]

isok(k) = if(k == 1, 1, my(f = factor(k)); vecmin(f[,2]) > 1 && !(numdiv(f) % (1<

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A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

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A382063 Numbers whose number of coreful divisors is divisible by their number of exponential divisors.

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A382062 Powerful numbers whose number of divisors is divisible by their number of unitary divisors.

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