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A189988 Numbers with prime factorization p^2*q^4.

%I A189988 #38 Feb 22 2025 01:51:25
%S A189988 144,324,400,784,1936,2025,2500,2704,3969,4624,5625,5776,8464,9604,
%T A189988 9801,13456,13689,15376,21609,21904,23409,26896,29241,29584,30625,
%U A189988 35344,42849,44944,55696,58564,59536,60025,68121,71824,75625,77841
%N A189988 Numbers with prime factorization p^2*q^4.
%C A189988 Numbers k such that tau(k^2)/tau(k) = 3 where tau(n) is the number of divisors of n (A000005). - _Michel Marcus_, Feb 09 2018
%H A189988 T. D. Noe, <a href="/A189988/b189988.txt">Table of n, a(n) for n = 1..1000</a>
%H A189988 Will Nicholes, <a href="https://web.archive.org/web/20180225065729/http://willnicholes.com/math/primesiglist.htm">Prime Signatures</a>
%H A189988 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A189988 Sum_{n>=1} 1/a(n) = P(2)*P(4) - P(6) = A085548 * A085964 - A085966 = 0.017749..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020
%F A189988 a(n) = A054753(n)^2. - _R. J. Mathar_, May 05 2023
%t A189988 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,4}; Select[Range[150000],f]
%t A189988 Module[{upto=80000},Select[Union[Flatten[{#[[1]]^2 #[[2]]^4,#[[1]]^4 #[[2]]^2}&/@ Subsets[Prime[Range[Sqrt[upto/16]]],{2}]]],#<=upto&]] (* _Harvey P. Dale_, Dec 15 2017 *)
%o A189988 (PARI) list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/4), t=p^4;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A189988 (Python)
%o A189988 from math import isqrt
%o A189988 from sympy import primepi, integer_nthroot, primerange
%o A189988 def A189988(n):
%o A189988     def bisection(f,kmin=0,kmax=1):
%o A189988         while f(kmax) > kmax: kmax <<= 1
%o A189988         kmin = kmax >> 1
%o A189988         while kmax-kmin > 1:
%o A189988             kmid = kmax+kmin>>1
%o A189988             if f(kmid) <= kmid:
%o A189988                 kmax = kmid
%o A189988             else:
%o A189988                 kmin = kmid
%o A189988         return kmax
%o A189988     def f(x): return n+x-sum(primepi(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])
%o A189988     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A189988 Cf. A178740.
%Y A189988 Cf. A085548, A085964, A085966.
%K A189988 nonn
%O A189988 1,1
%A A189988 _Vladimir Joseph Stephan Orlovsky_, May 03 2011