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

Author

Amiram Eldar, Jan 11 2019

Keywords

• nonn

Comments

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

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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 *)

Python

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