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

%I A323332 #21 May 06 2025 09:31:43
%S A323332 1,6,12,12,24,30,36,48,72,56,96,144,108,180,216,132,150,192,288,182,
%T A323332 336,360,432,360,324,384,576,306,648,392,380,672,720,864,672,792,900,
%U A323332 768,552,1152,750,1296,1080,1092,972,1344,1440,870,1728,2160,992,1584
%N A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).
%C A323332 The sum of the reciprocals of all the terms of this sequence is Pi^2/6 (A013661).
%C A323332 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
%H A323332 Amiram Eldar, <a href="/A323332/b323332.txt">Table of n, a(n) for n = 1..10000</a>
%H A323332 Eckford Cohen, <a href="https://doi.org/10.1090/S0002-9939-1961-0132717-6">A property of Dedekind's psi-function</a>, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.
%t A323332 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 *)
%o A323332 (Python)
%o A323332 from math import isqrt, prod
%o A323332 from sympy import mobius, integer_nthroot, primefactors
%o A323332 def A323332(n):
%o A323332     def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A323332     def bisection(f,kmin=0,kmax=1):
%o A323332         while f(kmax) > kmax: kmax <<= 1
%o A323332         while kmax-kmin > 1:
%o A323332             kmid = kmax+kmin>>1
%o A323332             if f(kmid) <= kmid:
%o A323332                 kmax = kmid
%o A323332             else:
%o A323332                 kmin = kmid
%o A323332         return kmax
%o A323332     def f(x):
%o A323332         c, l = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0
%o A323332         j = isqrt(x)
%o A323332         while j>1:
%o A323332             k2 = integer_nthroot(x//j**2,3)[0]+1
%o A323332             w = squarefreepi(k2-1)
%o A323332             c -= j*(w-l)
%o A323332             l, j = w, isqrt(x//k2**3)
%o A323332         return c+l
%o A323332     a = primefactors(m:=bisection(f,n,n))
%o A323332     return m*prod(p+1 for p in a)//prod(a) # _Chai Wah Wu_, Sep 14 2024
%Y A323332 Cf. A001615, A001694, A013661, A082695, A112526.
%Y A323332 Sequences whose density can be calculated using this sequence: A054979, A129575, A307958, A308053, A321147, A322858, A323310, A328135, A339936, A340109, A364990, A382061, A383693, A383695, A383697.
%K A323332 nonn
%O A323332 1,2
%A A323332 _Amiram Eldar_, Jan 11 2019