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A189987 Numbers with prime factorization p*q^6.

%I A189987 #34 Feb 22 2025 09:57:18
%S A189987 192,320,448,704,832,1088,1216,1458,1472,1856,1984,2368,2624,2752,
%T A189987 3008,3392,3645,3776,3904,4288,4544,4672,5056,5103,5312,5696,6208,
%U A189987 6464,6592,6848,6976,7232,8019,8128,8384,8768,8896,9477,9536,9664,10048,10432,10688
%N A189987 Numbers with prime factorization p*q^6.
%H A189987 T. D. Noe, <a href="/A189987/b189987.txt">Table of n, a(n) for n = 1..1000</a>
%H A189987 Will Nicholes, <a href="https://web.archive.org/web/20220628070358/http://willnicholes.com/math/primesiglist.htm">Prime Signatures</a>
%H A189987 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%t A189987 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,6}; Select[Range[30000],f]
%o A189987 (PARI) list(lim)=my(v=List(),t);forprime(p=2, (lim\2)^(1/6), t=p^6;forprime(q=2, lim\t, if(p==q, next);listput(v,t*q))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A189987 (Python)
%o A189987 from sympy import primepi, primerange, integer_nthroot
%o A189987 def A189987(n):
%o A189987     def bisection(f,kmin=0,kmax=1):
%o A189987         while f(kmax) > kmax: kmax <<= 1
%o A189987         kmin = kmax >> 1
%o A189987         while kmax-kmin > 1:
%o A189987             kmid = kmax+kmin>>1
%o A189987             if f(kmid) <= kmid:
%o A189987                 kmax = kmid
%o A189987             else:
%o A189987                 kmin = kmid
%o A189987         return kmax
%o A189987     def f(x): return n+x-sum(primepi(x//p**6) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,7)[0])
%o A189987     return bisection(f,n,n) # _Chai Wah Wu_, Feb 22 2025
%Y A189987 Cf. A030632, A092759.
%K A189987 nonn
%O A189987 1,1
%A A189987 _Vladimir Joseph Stephan Orlovsky_, May 03 2011